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# Finding a particular solution of a differential equation

I already solved the homogeneous portion, and I need help solving the particular solution and of course combining the two to get the entire solution to the differential equation. Not too difficult - see attachment. Please use equation editor if possible. Thank you.
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Given that:

dMS/dt = m(MN - MS) - pMS¬
MN¬ (0) = MS (0) = 0
And using:
M¬N + MS = [Po/(r + p)](ert - e-pt)
Show that

MS(t) =

BY SOLVING THE DIFFERENTIAL EQUATION FOR dM¬s¬/dt with a homogeneous and particular solution.

Using the MN + MS equation to solve for MN¬ in terms of MS gives the differential equation to be solved as:

dMS/dt + (2m+p)MS = [mPo/(r+p)]ert - [mPo/(r+p)]e-pt

I have already solved the homogeneous solution to this differential equation to get that :

MS = C1e-(2m+p)t where C1 is the constant of integration.

What I need is the particular solution, and the hint I received in class was to start by assuming that MS = C2ert + C4e-pt

And of course the solution will be the addition of the homogeneous and particular solutions, using the boundary conditions given above that MS¬(0)=MN(0) = 0
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(See attached file for full problem description and equations)

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The solution finds a particular solution of a differential equation.

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