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Chapter 1: Q5.3-13E (page 1)

In Problems 10鈥�13, use the vectorized Euler method with h = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

y''=t2-y2;y(0)=0,y'(0)=1on[0,1]

Short Answer

Expert verified

The required solution is;

y(0.25)=0.25y(0.5)=0.5y(0.75)=0.75y(1)=1

Step by step solution

01

Transform the equation

Here h=0.25 on [0,1].

The equations can be written as;

x1(t)=y(t)x2(t)=y'(t)=x'1

The transformation of the equation is;

x'1(t)=x2(t)x'2(t)=t2-x12

The initial conditions are;

x1(0)=y1(0)=0=x1,0x2(0)=y'(0)=1=x2,0

02

Apply Euler’s method

Now,

xn+1=xn+hf(tn,xn)

tn+1=tn+ht1=0+0.25=0.25x1(0.25)=x1,1=0.25x2(0.25)=x2,1=1

And

tn+1=tn+ht2=0.25+0.25=0.5x1(0.5)=x1,2=0.5x2(0.5)=x2,2=1

t3=0.5+0.25=0.75x1(0.75)=x1,3=0.75x2(0.75)=x2,3=1

t4=0.75+0.25=1x1(1)=x1,4=1x2(1)=x2,4=1

This is the required result.

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Most popular questions from this chapter

In problems 1-4 Use Euler鈥檚 method to approximate the solution to the given initial value problem at the points x=0.1,0.2,0.3,0.4, and 0.5, using steps of size 0.1h=0.1.

dydx=xy鈥�鈥�,鈥�鈥�y(0)=-1

Combat Model.A simplified mathematical model for conventional versus guerrilla combat is given by the systemx'1=-(0.1)x1x2;鈥�鈥�x1(0)=10;鈥�鈥�x'2=-x1;x2(0)=15 wherex1 andx2 are the strengths of guerrilla and conventional troops, respectively, and 0.1 and 1 are the combat effectiveness coefficients.Who will win the conflict: the conventional troops or the guerrillas? [Hint:Use the vectorized Runge鈥揔utta algorithm for systems with h=0.1to approximate the solutions.]

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

xd2ydx2+3x-dydx=e3x

Question:(a) Use the general solution given in Example 5 to solve the IVP. 4x"+e-0.1tx=0,x(0)=1,x'(0)=-12.Also use J'0(x)=-J1(x) and Y'0(x)=-Y1(x)=-Y1(x)along withTable 6.4.1 or a CAS to evaluate coefficients.

(b) Use a CAS to graph the solution obtained in part (a) for.

In Project C of Chapter 4, it was shown that the simple pendulum equation胃''(t)+蝉颈苍胃(t)=0 has periodic solutions when the initial displacement and velocity show that the period of the solution may depend on the initial conditions by using the vectorized Runge鈥揔utta algorithm with h= 0.02 to approximate the solutions to the simple pendulum problem on

[0, 4] for the initial conditions:

localid="1664100454791" (a)鈥�鈥�胃(0)=0.1,胃'(0)=0(b)鈥�鈥�胃(0)=0.5,胃'(0)=0(c)鈥�鈥�胃(0)=1.0,胃'(0)=0

[Hint: Approximate the length of time it takes to reach].
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