Question

Apply Euler’s method to approximate y(3) to the differential equation dy dx = x − y, y(0) = 1 using step size h=1.

Answer 1

So y(3)=1

Explanation:

Given that

`(dy)/(dx)=x-y`

y(0)=1,step size h=1

From Euler's method

`(dy)/(dx)=f(x,y)=x-y`

`y_(n+1)=y_n+hf(x_n,y_n),x_n=x_0+nh`

`y_1=y_0+hf(x_0,y_0)`

`y_1=1+1f(0,1)`

f(0,1)=0-1= -1

`y_1=1-1`=0

`y_(2)=y_1+hf(x_1,y_1)`

`y_(2)=0+1f(1,0)`

f(1,0)=1

`y_(2)=1`

`y_(3)=y_2+hf(x_2,y_2)`

`y_(3)=1+1f(2,1)`

f(2,1)=1

`y_(3)=1+1`=2

`y_(4)=y_3+hf(x_3,y_3)`

`y_(4)=2+1f(3,2)`

f(3,2)= -1

`y_(4)=2-1`=1

So y(3)=1