   Chapter 6.1, Problem 80E

Chapter
Section
Textbook Problem

Euler's Method In Exercises 79-81, complete the table using the exact solution of the differential equation and two approximations obtained using Euler’s Method to approximate the particular solution of the differential equation. Use h = 0.2 and h = 0.1 , and compute each approximation to four decimal places. x 0 0.2 0.4 0.6 0.8 1 y(x) (exact) y ( x ) ( h = 0.2 ) y ( x ) ( h = 0.1 ) Differential Equation Initial Exact Condition Solution d y d x = 2 x y ( 0.2 ) y = 2 x 2 + 4

To determine

To Calculate: Complete the table by using the exact solution of the differential equation, dydx=2xy and the two approximations obtained using Euler’s method.

Explanation

Given:

The values h=0.2 and h=0.1 along with the differential dydx=2xy. The initial condition is (0,2) with exact solution y=2x2+4.

Formula Used:

Euler’s method, xn=xn1+h and yn=yn1+hF(xn1,yn1).

Calculation:

Exact solution of differential equation,

y=2x2+4

The first two values of the exact solution of differential equation are

Putting x=0, we get that,

y=2x2+4=0+4=2

And now, putting x=0.2, we get that

y=2x2+4=2(0.2)2+4=2.0199

The table is

 x 0 0.2 0.4 0.6 0.8 1 y(x)(exact) 2 2.0199 2.0785 2.1726 2.2978 2.4495

Now, let us consider the given differential equation,

dydx=2xy

So, by the use of h=0.2,x0=0,y0=2, and F(x,y)=2xy, we get,

x0=0, x1=0.2, x2=0.4,x3=0.6, x4=0.8 and x5=1.0.

The first three approximations are

First approximation:

y1=y0+hF(x0,y0)=2+0.2F(0,2)=2+0.2{2(0)2}=2

Second approximation:

y2=y1+hF(x1,y1)=2+0.2F(0.2,2)=2+0.2{2(0.1)2}=2

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