Suppose \(p,q,f\) are continuous functions on an open interval \(I\). If \(y_1\) and \(y_2\) are linearly independent solutions to \(y^{\,\prime \prime }+py^{\,\prime }+qy=0\) then there is a particular solution to \(y^{\,\prime \prime }+py^{\,\prime }+qy=f\) of the form \(y=uy_1+vy_2\) where \(\displaystyle u^{\,\prime }=-\frac {fy_2}{W}\) and \(\displaystyle v^{\,\prime }=\frac {fy_1}{W}\) where \(W=y_1y_2^{\,\prime }-y_1^{\,\prime }y_2\) is the Wronskian of \(y_1\) and \(y_2\).

(a)
For each differential equation, use Variation of Parameters to find a particular solution. Then find the solution involving the least number of terms.
(i)
\(y^{\,\prime \prime }+y=\dfrac {\sin {(t)}}{\cos ^2{(t)}}\)
(ii)
\(x^2 y^{\,\prime \prime }-3xy^{\,\prime }+3y=2x^4 \sin {(x)}\)
(iii)
\((x-1)y^{\,\prime \prime }-xy^{\,\prime }+y=2(x-1)^2e^x\) given that \(y_1=x\) and \(y_2=e^x\) are complementary solutions.
(b)
Find the solution to the IVP \(x^2 y^{\,\prime \prime }+2xy^{\,\prime }-2y=9x\), \(y(1)=0\), \(y^{\,\prime }(1)=6\) on the interval \((0,\infty )\). Put the answer in the box provided.

\(y=\)
(c)
Find the solution to the IVP
\(4x y^{\,\prime \prime }+2y^{\,\prime }+y=\sin {(\sqrt {x})}\), \(y(\pi ^2)=\frac {\pi }{2}\), \(y^{\,\prime }(\pi ^2)=0\) on the interval \((0,\infty )\), given that \(y_1=\cos {(\sqrt {x})}\) and \(y_2=\sin {(\sqrt {x})}\) are complementary solutions. Put the answer in the box provided.

\(y=\)