Internal problem ID [250]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number: 55.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+4 y-\left (\sin ^{2}\relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 31
dsolve(diff(y(x),x$2)+4*y(x)=sin(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \sin \left (2 x \right ) c_{2}+\cos \left (2 x \right ) c_{1}-\frac {\sin \left (2 x \right ) x}{8}+\frac {1}{8}-\frac {\cos \left (2 x \right )}{8} \]
✓ Solution by Mathematica
Time used: 0.081 (sec). Leaf size: 63
DSolve[y''[x]+4*y[x]==sin[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sin (2 x) \left (\int _1^x\frac {1}{2} \cos (2 K[2]) \sin (K[2])^2dK[2]+c_2\right )+\cos (2 x) \left (\int _1^x-\cos (K[1]) \sin (K[1])^2 \sin (K[1])dK[1]+c_1\right ) \\ \end{align*}