Solve Process Engineering ODE without Laplace Transforms | ODE Help

[missmaria]

So this question pertains to some process engineering homework i have, which is basically the following:

I have an ODE that has the form: \frac{dh^{p}}{dt}+h^{p}+\int h^{p}=F^{p}{t}
where F^{p}{t} is the unit ramp function (i.e. F^{p}{t}=0 when t<0, and is equal to t when F^{p}\geq0

So my question is how do i solve this ODE without using Laplace transforms?? Can there even be an integral in such an ODE??

Thanks in advance, i appreciate your help

 

[matematikawan]

We can remove the integral sign by differentiating the equation with respect to t

h" + h' + h = F'(t)

This is a nonhomegeneous linear DE with constant coefficients which can be solved quite easily without using the Laplace transform.