Sketch the response of the following 1st order system to a r

Sketch the response of the following 1st order system to a ramp input with r=0.1. 5dx(t)/dt+2x(t)=2*rt and x(0)=0

Solution

5dx(t)/dt + 2 x(t) = 2rt and x(0) = 0

dx(t)/dt + 2/5 x(t) = 2/5 rt

This is of the form of dy/dx + Py = Q, Solution is y(IF) = (IF) Q dx

Here dx/dt + 2/5 x = 2/5 0.1 t

IF = e^2/5dt = e ^2/5t

So the solution is

x( e ^2/5t ) =  e ^2/5t 2/5 rt dt = 2/5r e ^2/5t t dt = 2/5 r [ 5/2 t e^2/5t - (5/2)² e^2/5t + c]

When t = 0 x = 0

Implies c = (5/2)²

So the solution is x( e ^2/5t ) == 2/5 r [ 5/2 t e^2/5t - (5/2)² e^2/5t + (5/2)² ]