Use Euler's method?

Question

Suppose dy/dx=10x/(x+y) and y=2 when x=0. Use Euler's method with two steps to estimate y at x=1. 

 

Answer: 3

 

Please show all your work.

Answer 1

For the differential equation y'=f(y,x), y(x₀)=y₀, Euler method looks as follows:

 

1) Suppose, we have initial value of y(x), here it is y(0)=2. Then, if we need to estimate y(1) in two steps, our step size is 0.5.

2) y_n+1=y_n+h*f(y_n,x_n)--general formula. In our case, h=0.5; f(y,x)=10x/(x+y)

 

First step: x₀=0, y₀=2; f(y₀,x₀)=10*0/(0+2)=0; y₁=y₀+0.5*f(y₀,x₀) =2+0.5*0=2;

Second step: x₁=0.5; y₁=2

f(y₁,x₁)=10*0.5/(0.5+2)=2; y₂=y₁+h*f(y₁,x₁)=2+0.5*2=3;

 

Answer: 3