Question:

a) Find a particular solution to \(x''+4x'+4x=8{{t}^{2}}+8\)

b) Find the solution to the following differential equation:

\[\frac{2ty}{{{t}^{2}}+1}-2t-\left( 2-\ln \left( {{t}^{2}}+1 \right) \right)y'=0\]

with y(5) = 0.

c) Solve \(tx'=4t-3x\)

d) Find the differential equation that has the solution to the complementary homogeneous differential equation of:

\[{{y}_{c}}\left( x \right)={{c}_{1}}{{e}^{-x}}\cos \left( 2x \right)+{{c}_{2}}{{e}^{-x}}\sin \left( 2x \right)\]

and a particular solution of the differential equation is:

\[{{y}_{p}}\left( x \right)=3\cos \left( 2x \right)-6\sin \left( 3x \right)\]