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Chapter 1: Q5.3-16E (page 1)

In Problems 14–24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge–Kutta algorithm. (At the instructor’s discretion, other algorithms may be used.)†

Using the vectorized Runge–Kutta algorithm for systems withh=0.175, approximate the solution to the initial value problemx'=2x-y;x(0)=0,y'=3x+6y;y(0)=-2 att=1.

Compare this approximation to the actual solution.

Short Answer

Expert verified

The solution isy1=-423.48and x1=127.77.

Step by step solution

01

Transform the equation

Write the equation as x'=2x-yand y'=3x+6y

The transformation of the equation is:

x'1t=x2tx1t=ytx2t=2x-x1x'2t=y'(tx'2t=3x+6x1t

The initial conditions are:

x(0)=0y(1)=-2

02

Apply Runge –Kutta method

For the solution, apply the Runge-Kutta method in MATLAB, and the solution isy1=-423.48 and x1=127.77.

03

Compare this approximation to the actual solution x(t)=e5t-e3t,y(t)=e3t-3e5t

By putting the value oft=1

x1=e5-e3=128.32y1=e3-3e5=-425.15

Therefore, the approximation solution isx1=128.32 and y1=-425.15.

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Most popular questions from this chapter

In Problems 9–13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

exy+y=x-1,dydx=e-xy-ye-xy+x

In Problems 14–24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge–Kutta algorithm. (At the instructor’s discretion, other algorithms may be used.)†

Using the vectorized Runge–Kutta algorithm, approximate the solution to the initial value problem y''=t2+y2;y(0)=1,y'(0)=0at t=1. Starting with h=1, continue halving the step size until two successive approximations of both y1andy'1differ by at most 0.1.

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

xd2ydx2+3x-dydx=e3x

Implicit Function Theorem. Let G(x,y)have continuous first partial derivatives in the rectangleR={x,y:a<x<b,c<y<d}containing the pointlocalid="1664009358887" (x0,y0). IfG(x0,y0)=0 and the partial derivativeGy(x0,y0)≠0, then there exists a differentiable function y=ϕ(x), defined in some intervalI=(x0-δ,y0+δ),that satisfies G for allforG(x,ϕx)all x∈I.

The implicit function theorem gives conditions under which the relationshipG(x,y)=0 implicitly defines yas a function of x. Use the implicit function theorem to show that the relationshipx+y+exy=0 given in Example 4, defines y implicitly as a function of x near the point(0,-1).

Question: Let f(x)and g(x)be analytic at x0. Determine whether the following statements are always true or sometimes false:

(a) 3f(x)+g(x)is analytic at x0.

(b) f(x)/g(x)is analytic at x0.

(c) f'(x)is analytic at x0.

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