Solving an ODE with a derivative, integral, and unit ramp function

[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$$^{p}$$$$\geq$$0

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.