Question

Find the solution of the given initial value problem. y'-y=11te^2t, y(0)=1

Answer 1

`y'-y=11te^(2t)`

`e^(-t)y'-e^(-t)y=11te^t`

`(\mathrm d)/(\mathrm dt)\left[e^(-t)y\right]=11te^t`

`e^(-t)y=11\displaystyle\int te^t\,\mathrm dt`

`e^(-t)y=11e^t(t-1)+C`

Since
`y(0)=1`, you have


`1=11(-1)+C\implies C=12`

and so


`e^(-t)y=11e^t(t-1)+12`

`y=11e^(2t)(t-2)+12e^t`