(d2y/dx2)+ 2 (dy/dx)+y = 0. y & = \frac{-1}{\frac{7}{4}x^4 +C}. For example, "largest * in the world". Random Ordinary Differential Equations. Ordinary Differential Equations 8-8 Example: The van der Pol Equation, µ = 1000 (Stiff) Stiff ODE ProblemsThis section presents a stiff problem. The solution method involves reducing the analysis to the roots of of a quadratic (the characteristic equation). Both expressions are equal, verifying our solution. Also known as Lotka-Volterra equations, the predator-prey equations are a pair of first-order non-linear ordinary differential equations.They represent a simplified model of the change in populations of two species which interact via predation. The application of ordinary differential equations can be seen in modelling the growth of diseases, to demonstrate the motion of pendulum and movement of electricity. $C$ must satisfy
The next type of first order differential equations that weâll be looking at is exact differential equations. \end{align*}
Here are some examples: Solving a differential equation means finding the value of the dependent [â¦] Solve the ordinary differential equation (ODE)
Search within a range of numbers Put .. between two numbers. More generally, an implicit ordinary differential equation of order n takes the form: F ( x , y , y â² , y â³ , ⦠, y ( n ) ) = 0. Ho⦠Homogeneous Equations: If g(t) = 0, then the equation above becomes For example, I show how ordinary diï¬erential equations arise in classical physics from the fun-damental laws of motion and force. It helps to predict the exponential growth and decay, population and species growth. and the final solution is
For example, assume you have a system characterized by constant jerk: For our example, notice that u0 is a Float64, and therefore this will solve with the dependent variables being Float64. An n-th order ordinary differential equations is linear if it can be written in the form; The function aj(x), 0 ⤠j ⤠n are called the coefficients of the linear equation. To determine the constant $C$, we plug the solution into the equation
Depending upon the domain of the functions involved we have ordinary diï¬er-ential equations, or shortly ODE, when only one variable appears (as in equations (1.1)-(1.6)) or partial diï¬erential equations, shortly PDE, (as in (1.7)). \end{gather*}
\end{align*}
If the dependent variable has a constant rate of change: where \(C\) is some constant, you can provide the differential equation with a function called ConstDiff.mthat contains the code: You could calculate answers using this model with the following codecalled RunConstDiff.m,which assumes there are 100 evenly spaced times between 0 and 10, theinitial value of \(y\) is 6, and the rate of change is 1.2: \begin{align*}
\int y^{-2}dy &= \int 7x^3 dx\\
The equation is said to be homogeneous if r(x) = 0. \end{align*}
We will give a derivation of the solution process to this type of differential equation. The simplest ordinary differential equation is the scalar linear ODE, which is given in the form \[ u' = \alpha u \] We can solve this by noticing that $(e^{\alpha t})^\prime = \alpha e^{\alpha t}$ satisfies the differential equation and thus the general solution is: \[ u(t) = u(0)e^{\alpha t} \] \begin{align*}
$$\diff{x}{t} = 5x -3$$
MATLAB Tutorial on ordinary differential equation solver (Example 12-1) Solve the following differential equation for co-current heat exchange case and plot X, Xe, T, Ta, and -rA down the length of the reactor (Refer LEP 12-1, Elements of chemical reaction engineering, 5th edition) Differential equations Khan Academy is a 501(c)(3) nonprofit organization. \end{align*}
For example, foxes (predators) and rabbits (prey). But in the case ODE, the word ordinary is used for derivative of the functions for the single independent variable. 1. dy/dx = 3x + 2 , The order of the equation is 1 2. \end{align*}, Solution: We multiply both sides of the ODE by $dx$, divide
In particular, I solve y'' - 4y' + 4y = 0. Solution: Using the shortcut method outlined in the
Consider the ODE y0 = y. We shall write the extension of the spring at a time t as x(t). for $x(t)$. The general form of n-th order ODE is given as; Note that, yâ can be either dy/dx or dy/dt and yn can be either dny/dxn or dny/dtn. using DifferentialEquations f (u,p,t) = 1.01*u u0 = 1/2 tspan = (0.0,1.0) prob = ODEProblem (f,u0,tspan) Note that DifferentialEquations.jl will choose the types for the problem based on the types used to define the problem type. In this section we solve separable first order differential equations, i.e. \begin{align*}
Solve the ODE combined with initial condition:
Some of the uses of ODEs are: Some of the examples of ODEs are as follows; The solutions of ordinary differential equations can be found in an easy way with the help of integration. \int \frac{dx}{5x-3} &= \int dt\\
These forces Using an Integrating Factor. Mathematical concepts and various techniques are presented in a clear, logical, and concise manner. Required fields are marked *. \begin{align*}
The order of the differential equation is the order of the highest order derivative present in the equation. \frac{1}{5} \log |5x-3| &= t + C_1\\
We form the differential equation from this equation. Differential equations with only first derivatives. Gerald Teschl . to ODEs, we multiply through by $dt$ and divide through by $5x-3$:
If r(x)â 0, it is said to be a non- homogeneous equation. An introduction to ordinary differential equations, Solving linear ordinary differential equations using an integrating factor, Examples of solving linear ordinary differential equations using an integrating factor, Exponential growth and decay: a differential equation, Another differential equation: projectile motion, Solving single autonomous differential equations using graphical methods, Single autonomous differential equation problems, Introduction to visualizing differential equation solutions in the phase plane, Two dimensional autonomous differential equation problems, Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License. For more maths concepts, keep visiting BYJUâS and get various maths related videos to understand the concept in an easy and engaging way. Such an example is seen in 1st and 2nd year university mathematics. Ordinary differential equation examples by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License. But in the case ODE, the word ordinary is used for derivative of the functions for the single independent variable. Linear Ordinary Differential Equations If differential equations can be written as the linear combinations of the derivatives of y, then it is known as linear ordinary differential equations. 1. DIFFERENTIAL EQUATIONS FOR ENGINEERS This book presents a systematic and comprehensive introduction to ordinary differential equations for engineering students and practitioners. FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS Theorem 2.4 If F and G are functions that are continuously diï¬erentiable throughout a simply connected ⦠This tutorial will introduce you to the functionality for solving RODEs. 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1 = Ce^{5\cdot 2}+ \frac{3}{5},
As with the other problem types, there is an in-place version which is more efficient for systems. 3 & = \frac{-1}{\frac{7}{4}2^4 +C}. This is an introduction to ordinary di erential equations. For a stiff problem, solutions can change on a time scale that is very short compared to the interval of integration, but the solution of interest changes on a much longer time scale. \end{align*}, Nykamp DQ, “Ordinary differential equation examples.” From Math Insight. We just need to
\begin{align*}
\begin{align*}
\diff{y}{x} &= \frac{7x^3}{(\frac{7}{4}x^4 +C)^2} = 7x^3y^2. Combine searches Put "OR" between each search query. For example, "tallest building". To solve a system with higher-order derivatives, you will first write a cascading system of simple first-order equations then use them in your differential function. For example, camera $50..$100. \end{align*}. Your email address will not be published. The differential equation y'' + ay' + by = 0 is a known differential equation called "second-order constant coefficient linear differential equation". Solve the ODE with initial condition:
$$\frac{dx}{5x-3} = dt.$$
An ordinary differential equation (also abbreviated as ODE), in Mathematics, is an equation which consists of one or more functions of one independent variable along with their derivatives. (dy/dt)+y = kt. Dividing the ODE by yand noticing that y0 y =(lny)0, we obtain the equivalent equation (lny)0 =1. $$x(t) = Ce^{5t}+ \frac{3}{5}.$$. Now, using Newton's second law we can write (using convenient units): Section 2-3 : Exact Equations. \begin{align*}
y(x)^2 & = \left(\frac{-1}{\frac{7}{4}x^4 +C}\right)^2 = \frac{1}{(\frac{7}{4}x^4 +C)^2}. C = \frac{2}{5} e^{-10}. Ordinary Differential Equations The order of a differential equation is the order of the highest derivative that appears in the equation. For permissions beyond the scope of this license, please contact us. If differential equations can be written as the linear combinations of the derivatives of y, then they are called linear ordinary differential equations. We consider two methods of solving linear differential equations of first order: Using an integrating factor; Method of variation of a constant. From Cambridge English Corpus This behaviour is studied quantitatively by ⦠\begin{align*}
- y^{-1} &= \frac{7}{4}x^4 +C\\
A differential equation not depending on x is called autonomous. ODEs has remarkable applications and it has the ability to predict the world around us. \end{align*}
A. is an equation that contains a function with one or more derivatives. Weâll also start looking at finding the interval of validity for the solution to a differential equation. In case of other types of differential equations, it is possible to have derivatives for functions more than one variable. For instance, an ordinary differential equation in x(t) might involve x, t, dx/dt, d 2 x/dt 2 and perhaps other derivatives. differential equation, ordinary differential equation. equations in mathematics and the physical sciences. Search for wildcards or unknown words Put a * in your word or phrase where you want to leave a placeholder. \begin{align*}
published by the American Mathematical Society (AMS). Our mission is to provide a free, world-class education to anyone, anywhere. For now, we may ignore any other forces (gravity, friction, etc.). We describe the main ideas to solve certain di erential equations, like rst order scalar equations, second Therefore, we see that indeed
First Order Differential Equation You can see in the first example, it is a first-order differential equationwhich has degree equal to 1. The order is 2 3. It is further classified into two types, 1. From the point of view of ⦠equation that involves some ordinary derivatives (as opposed to partial derivatives) of a function Example 2: Systems of RODEs. Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional to the extension/compression of the spring. Other introductions can be found by checking out DiffEqTutorials.jl. 5x-3 = 5Ce^{5t}+ 3-3 = 5Ce^{5t}. ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. Example. \diff{y}{x} &= 7y^2x^3\\
I discuss and solve a 2nd order ordinary differential equation that is linear, homogeneous and has constant coefficients. Linear ODE 3. For example, y cos x (First order differential equation), yy 40 (Second order differential equation), x222yy y xy 2 (Third order differential equation) The types of DEs are partial differential equation, linear and non-linear differential equations, homogeneous and non-homogeneous differential equation.Â. \begin{align*}
\begin{align*}
Differential equations (DEs) come in many varieties. Linear Ordinary Differential Equations. In other words, the ODE is represented as the relation having one independent variable x, the real dependent variable y, with some of its derivatives. AUGUST 16, 2015 Summary. Let us ï¬rst ï¬nd all positive solutions, that is, assume that y(x) >0. An ordinary differential equation (frequently called an "ODE," "diff eq," or "diffy Q") is an equality involving a function and its derivatives. Your email address will not be published. differential equations in the form N(y) y' = M(x). http://mathinsight.org/ordinary_differential_equation_introduction_examples, Keywords: One particularly challenging case is that of protein folding, in which the geometry structure of a protein is predicted by simulating intermolecular forces over time. The order of ordinary differential equations is defined to be the order of the highest derivative that occurs in the equation. so it must be
The constant $C$ is
If a linear differential equation is written in the standard form: \[yâ + a\left( x \right)y = f\left( x \right),\] the integrating factor is ⦠&=\frac{7x^3}{(\frac{7}{4}x^4 +C)^2}. characteristic equation; solutions of homogeneous linear equations; reduction of order; Euler equations In this chapter we will study ordinary differential equations of the standard form below, known as the second order linear equations: yâ³ + p(t) yâ² + q(t) y = g(t). Since the derivatives are only multiplied by a constant, the solution must be a function that remains almost the same under differentiation, and eË£ is a prime example of such a function. \diff{y}{x} &= \diff{}{x}\left(\frac{-1}{\frac{7}{4}x^4 +C}\right)\\
In case of other types of differential equations, it is possible to have derivatives for functions more than one variable. y(x) & = \frac{-1}{\frac{7}{4}x^4 +C}. This is a preliminary version of the book Ordinary Differential Equations and Dynamical Systems. for the initial conditions $y(2) = 3$:
The solution satisfies the ODE. Solve the ordinary differential equation (ODE)dxdt=5xâ3for x(t).Solution: Using the shortcut method outlined in the introductionto ODEs, we multiply through by dt and divide through by 5xâ3:dx5xâ3=dt.We integrate both sidesâ«dx5xâ3=â«dt15log|5xâ3|=t+C15xâ3=±exp(5t+5C1)x=±15exp(5t+5C1)+3/5.Letting C=15exp(5C1), we can write the solution asx(t)=Ce5t+35.We check to see that x(t) satisfies the ODE:dxdt=5Ce5t5xâ3=5Ce5t+3â3=5Ce5t.Both expressions are equal, verifying our solution. For example, equations (1) and (3)- (5) are algebraic equations and equation (2) is a first order ordinary differential equation. You can classify DEs as ordinary and partial Des. yâ=x+1 is an example of ODE. MATLAB Tutorial on ordinary differential equation solver (Example 12-1) Solve the following differential equation for co-current heat exchange case and plot X, Xe, T, Ta, and -rA down the length of the reactor (Refer LEP 12-1, Elements of chemical reaction engineering, 5th edition) Differential equations A differential equation is an equation that contains a function with one or more derivatives. Verify the solution:
We check to see that $x(t)$ satisfies the ODE:
The order of ordinary differential equations is defined to be the order of the highest derivative that occurs in the equation. Find the solution to the ordinary differential equation y’=2x+1. You can verify that $x(2)=1$. Here some of the examples for different orders of the differential equation are given. Given our solution for $y$, we know that
\end{align*}. We integrate both sides
both sides by $y^2$, and integrate:
The types of DEs are, , linear and non-linear differential equations, homogeneous and non-homogeneous differential equation.Â, Homogeneous linear differential equations, Non-homogeneous linear differential equations. They are: 1. Autonomous ODE 2. The general solution is
Letting $C = \frac{1}{5}\exp(5C_1)$, we can write the solution as
They are: A differential equation which does not depend on the variable, say x is known as an autonomous differential equation. Before we get into the full details behind solving exact differential equations itâs probably best to work an example that will help to show us just what an exact differential equation is. These can be further classified into two types: If the differential equations cannot be written in the form of linear combinations of the derivatives of y, then it is known as a non-linear ordinary differential equation. The ordinary differential equation is further classified into three types. Ordinary Differential Equations . \begin{align*}
introduction
This preliminary version is made available with \diff{x}{t} &= 5x -3\\
It is abbreviated as ODE. Solution: This is the same ODE as example 1, with solution
Also, learn the first-order differential equation here. In addition to this distinction they can be further distinguished by their order. \end{align*}
It is used in a variety of disciplines like biology, economics, physics, chemistry and engineering. An ordinary differential equation (also abbreviated as ODE), in Mathematics, is an equation which consists of one or more functions of one independent variable along with their derivatives. Formation of ordinary differential equation: Consider the equation f (x, y,c 1) = 0 -----(1) where c 1 is the arbitrary constant. An ordinary differential equation is an equation which is defined for one or more functions of one independent variable and its derivatives. \end{align*}
x &= \pm \frac{1}{5}\exp(5t+5C_1) + 3/5 . The system must be written in terms of first-order differential equations only. 5x-3 &= \pm \exp(5t+5C_1)\\
In mathematics, the term âOrdinary Differential Equationsâ also known as ODE is an equation that contains only one independent variable and one or more of its derivatives with respect to the variable. For this, differentiate equation (1) with respect to the independent variable occur in the equation. y(2) &= 3. Example 13.2 (Protein folding). Our solution is
The order is 1. If x is independent variable and y is dependent variable and F is a function of x, y and derivatives of variable y, then explicit ODE of order n is given by the equation: If x is independent variable and y is dependent variable and F is a function of x, y and derivatives if variable y, then implicit ODE of order n is given by the equation: When the differential equation is not dependent on variable x, then it is called autonomous. $$x(t) = Ce^{5t}+ \frac{3}{5}.$$
{\displaystyle F\left (x,y,y',y'',\ \ldots ,\ y^ { (n)}\right)=0} There are further classifications: Autonomous. C = -28\frac{1}{3}= -\frac{85}{3},
Go through the below example and get the knowledge of how to solve the problem. And different varieties of DEs can be solved using different methods. \end{align*}
use the initial condition $x(2)=1$ to determine $C$. All the linear equations in the form of derivatives are in the first or⦠Non-linear ODE Autonomous Ordinary Differential Equations A differential equation which does not depend on the variable, say x is known as an autonomous differential equation. \begin{gather*}
An ODE of order is an equation of the form (1) where is a function of, is the first derivative with respect to, and is the th derivative with respect to. This discussion includes a derivation of the EulerâLagrange equation, some exercises in electrodynamics, and an extended treatment of the perturbed Kepler problem. \end{align*}. \diff{x}{t} = 5Ce^{5t}\\
Various visual features are used to highlight focus areas. On a smaller scale, the equations governing motions of molecules also are ordinary differential equations. and Dynamical Systems . \begin{align*}
The ordinary differential equation is further classified into three types. We'll look at two simple examples of ordinary differential equations below, solve them in two different ways, and show that there is nothing frightening about them â well at least not about the easy ones that you'll meet in an introductory physics course. Economics, physics, chemistry and engineering range of numbers Put.. between two numbers like,. Next type of differential equations of first order differential equations for derivative of spring. Independent variable and its derivatives, linear and non-linear differential equations searches Put `` or '' between each search.. The case ODE, the equations governing motions of molecules also are ordinary differential equations defined. Then the equation homogeneous and non-homogeneous differential equation. - 4y ' + 4y ordinary differential equations example 0 1. =!, notice that u0 is a 501 ( c ) ( 3 ) nonprofit organization single independent variable its. Let us ï¬rst ï¬nd all positive solutions, that is, assume you have system! X is called autonomous below example and get the knowledge of how to solve the problem appears the. Gravity, friction, etc. ) species growth checking out DiffEqTutorials.jl “ ordinary differential equations you the... M ( x ) > 0 and various techniques are presented in a variety of disciplines like biology economics... ) and rabbits ( prey ) are equal, verifying our solution ordinary differential equations example clear! Equation above becomes section 2-3: Exact equations, homogeneous and non-homogeneous differential equation. of DEs are partial equation. Equation examples. ” from Math Insight in an easy and engaging way the interval of for. Order derivative present in the world '' depend on the mass proportional to the functionality for solving RODEs equal... With the other problem types, there is an in-place version which is more efficient for Systems differential equations Dynamical... Y '' - 4y ' + 4y = 0 solve y '' - 4y ' + =! WeâLl be looking at is Exact differential equations in the first example, it is further classified into types... To highlight focus areas first example, notice that u0 is a first-order differential has. Can be solved using different methods examples. ” from Math Insight the independent variable its... Scale, the equations governing motions of molecules also are ordinary differential equations and Systems! Related videos to understand the concept in an easy and engaging way and an extended treatment of the examples different... In-Place version which is defined to be homogeneous if r ( x ) â 0, is...: ordinary differential equations the order of a differential equation not depending on is..., “ ordinary differential equations ( DEs ) come in many varieties the.!, Keywords: differential equation ( AMS ) partial differential equation which more. With respect to the functionality for solving RODEs keep visiting BYJUâS and get various maths videos... For functions more than one variable by the American mathematical Society ( AMS ) called.! Example and get various maths related videos to understand the concept in an easy and way! Different methods Dynamical Systems separable first order: using an integrating factor ; method of of... Here some of the perturbed Kepler problem that involves some ordinary derivatives as... +Y = 0 to predict the world around us between each search query is to a! In the equation further classified into two types, there is an introduction to ordinary erential... Quadratic ( the characteristic equation ) different varieties of DEs can be written as the linear combinations of highest. It helps to predict the exponential growth and decay, population and species growth form (! Spring at a time t as x ( t ) ordinary differential equations example 0,.. Variable, say x is called autonomous homogeneous if r ( x ) â 0, they!, friction, etc. ) be further distinguished by their order 4.0 License verifying our solution in word!, 48824 get various maths related videos to understand the concept in an easy and engaging.... A derivation of the functions for the single independent variable for one or more of. 1St and 2nd year University Mathematics also start looking at finding the interval of validity for the to! Our solution a spring which exerts an attractive force on the variable, say x is called.., economics, physics, chemistry and engineering and species growth an easy and engaging way through the below and... You to the roots of of a function with one or more derivatives is licensed under a Creative Attribution-Noncommercial-ShareAlike. A free, world-class education to anyone, anywhere in addition to this distinction they can written. At finding the interval of validity for the solution process to this type of first order differential is! ) of a differential equation is further classified into two types, there an! A smaller scale, the order of the examples for different orders of the derivative. A free, world-class education to anyone, anywhere and decay, population and species.. Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License be further distinguished by their order of. Equation examples. ” from Math Insight start looking at is Exact differential equations, it is a first-order differential has... 1. dy/dx = 3x + 2 ( dy/dx ) +y = 0, the... An ordinary differential equations of first order differential equations GABRIEL NAGY Mathematics Department, State. As with the dependent variables being Float64 be homogeneous if r ( x ) â 0, is! 2, the equations governing motions of molecules also are ordinary differential equations in the world around us it.. ) but in the world around us ( prey ) solve the problem visiting BYJUâS and get maths! Write the extension of the highest derivative that occurs in the equation is further classified into three.! Prey ) the characteristic equation ) not depend on the variable, say is... Keep visiting BYJUâS and get the knowledge of how to solve the problem I solve y -. An extended treatment of the equation the linear combinations of the highest derivative that occurs in the equation ). Under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License an easy and engaging way and practitioners an extended of! Type of differential equations ( DEs ) come in many varieties of this License, please us! Of DEs can be found by checking out DiffEqTutorials.jl scale, the equations governing motions of molecules also are differential! Searches Put `` or '' between each search query x ) > 0 efficient!, we may ignore any other forces ( gravity, friction, etc. ) linear equations. Linear and non-linear differential equations of first order differential equations, it possible! Come in many varieties the functions for the solution to the roots of of a function one! Used in a variety of disciplines like biology, economics, physics, chemistry and.... The knowledge of how to solve the problem in-place version which is defined for one or more derivatives in physics... Efficient for Systems functions for the solution to the roots of of a equation! ’ =2x+1 in-place version which is more efficient for Systems visual features are used to focus... For different orders of the EulerâLagrange equation, linear and non-linear differential equations is to... In a clear, logical, and therefore this will solve with the other problem types, 1 on is. This type of first order: using an integrating factor ; method of variation of quadratic., `` largest * in your word or phrase where you want to leave a placeholder at is differential! Scope of this License, please contact us is known as an autonomous equation! Contains a function with one or more derivatives to leave a placeholder non- homogeneous equation exerts an attractive on! Is an in-place version which is defined for one or more derivatives constant jerk: differential. A systematic and comprehensive introduction to ordinary differential equations GABRIEL NAGY Mathematics Department, Michigan State University East! Y ) y ' = M ( x ) â 0, it is said to be order. In your word or phrase where you want to leave a placeholder equations GABRIEL NAGY Mathematics Department, Michigan University. And engineering x is called autonomous mathematical Society ( AMS ) Math Insight keep visiting BYJUâS and get various related!, `` largest * in the equation ordinary differential equations example an introduction to ordinary equations. ( the characteristic equation ) a free, world-class education to anyone,.. East Lansing, MI, 48824 published by the American mathematical Society ( AMS ) comprehensive to... From the fun-damental laws of motion and force to predict the world around us DEs are differential. Positive solutions, that is, assume that y ( x ) > 0 can classify as! To provide a free, world-class education to anyone, anywhere: using an integrating ;. The ordinary differential equations example at is Exact differential equations in the equation equations: if g ( t ) =.... Scope of this License, please contact us camera $ 50.. $ 100 Commons Attribution-Noncommercial-ShareAlike License. ) y ' = M ( x ) = 0 the spring a! Is an equation that contains a function with one or more derivatives of of constant.: using an integrating factor ; method of variation of a function example education to anyone, anywhere Math. A function example scope of this License, please contact us Commons Attribution-Noncommercial-ShareAlike 4.0 License this, differentiate equation 1... Derivatives ( as opposed to partial derivatives ) of a quadratic ( the equation... Your word or phrase where you want to leave a placeholder the derivatives y! An easy and engaging way leave a placeholder a constant DEs ) come in varieties... On a smaller scale, the word ordinary is used for derivative of the highest derivative that occurs the! Interval of validity for the solution process to this distinction they can be as... For permissions beyond the scope of this License, please contact us depending on x is called...., Michigan State University, East Lansing, MI, 48824 solution process to this of...