Question

Any help with this greatly appreciated.

Answer 1

Put the equation in standard linear form.

`x'(t) + (x(t))/(t + 5) = 5e^(5t)`

Find the integrating factor.

`\mu = \exp\left(\displaystyle \int (dt)/(t+5)\right) = e^(\ln|t+5|) = t+5`

Multiply both sides by
`\mu`.

`(t+5) x'(t) + x(t) = 5(t+5)e^(5t)`

Now the left side the derivative of a product,

`\bigg((t+5) x(t)\bigg)' = 5(t+5)e^(5t)`

Integrate both sides.

`(t+5) x(t) = \displaystyle 5 \int (t+5) e^(5t) \, dt`

On the right side, integrate by parts.

`(t+5) x(t) = \frac15 (5t+24) e^(5t) + C`

Solve for
`x(t)`.

`\boxed{x(t) = (5t+24)/(5t+25) e^(5t) + \frac C{t+5}}`