Question

Consider the differential equation dy/dx = xy. Let y = f(x) be the function that satisfies the differential equation with initial condition f(1) = 1. Use Euler's Method, starting at x = 1 with a step size of .1, to approximate f(1.2).

Answer 1

`f(1.2)=1.221`

Explanation:

From the question we are told that

`dy/dx = xy`

Initial condition
`f(1) = 1.`

`x = 1`

`Step\ Size\ of\ 0.1`

Generally equation for
`y_1` is given mathematically as

`f_0=F(1,1)=1`

Therefore

`y_1=y_0+hf_0`

`y_1=y_0+(0.1)(1)`

`y_1=1.1`

Generally the approximation of the solution is mathematically given by

`x=1+h\\x=1+0.1\\x=1.1`

`y=1.1`

Generally the
`f_1` is mathematically given by

`F_1=f(1.1,1.1)\\F_1=1.21`

Generally the
`y_2` is mathematically given by

`y_2=y_1+hf_1\\y_2=1.1+0.121\\y_2=1.221`

Generally the approximation to the solution

at

`y_2=1.221`

`x_2=1.1+0.1\\x_2=1.2`

Therefore

`f(1.2)=1.221`