Question

Problem1 The behavior of a physical system can be described by the following first order differential equation: dy/dt=2y +t^2

Answer 1

`y = (t^3e^(2t))/(3)+Ce^(2t)`.

Explanation:

Using first order linear differential equation:

`\frac{\mathrm{d} y}{\mathrm{d} t} = 2y + t^2`

`\frac{\mathrm{d} y}{\mathrm{d} t} - 2y =t^2`

finding integrating factor:

I.F =
`e^(\int -2dt)`

I.F =
`e^(-2t)`

now,

`y = (1)/(IF)(\int t^2dt+ c )`

`y = (1)/(e^(-2t))(\int t^2dt+ c )`

`y = (t^3e^(2t))/(3)+Ce^(2t)`

hence the solution is

`y = (t^3e^(2t))/(3)+Ce^(2t)`