Question

Find an explicit solution (solved for y) of the given initial-value problem in terms of an integral function. dy/dx + 3y = e^x^5, y(2) = 5.

Answer 1

Explanation:

Using linear differential equation method:

\frac{\mathrm{d} y}{\mathrm{d} x}+3y=e^5^x

I.F.=
`e^{\int {Q} \, dx }`

I.F.=
`e^{\int {3} \, dx }`

I.F.=
`e^(3x)`

y(x)=
`(1)/(e^(3x))[\int {e^(5x)} \, dx+c]`

y(x)=
`(e^(2x))/(5)+e^(-3x)* c`

substituting x=2

c=
`(25-e^4)/(5e^(-6))`

Now

y=
`(e^(2x))/(5)+e^(-3x)* (25-e^4)/(5e^(-6))`