Find the solution the the differential equation d^2y/dx^2 + (3/2)dy/dx + y = 22e^(-4x)
We first find the complementary function by guessing y=e^(kx). Substituting this into the equation d^2y/dx^2 + (3/2)dy/dx + y = 0. we find k^2 + (3/2)k + 1 = 0 which factorises into (k+2)(k+1/2). So our complementary function is y= Ae^(-2x) + Be^(-x/2). Now we find any particular integral by guessing y = Le^(-4x). Substituting this in to the equation d^2y/dx^2 + (3/2)dy/dx + y = 22e^(-4x) we find that L(16e^(-4x) - 4e^(-4x) + e^(-4x)) = 22e^(-4x) and L=2. So the solution to the differential equation is y= Ae^(-2x) + Be^(-x/2) + 2e^(-4x) //
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