8.11.5 problem 5

Internal problem ID [873]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 5
Date solved : Wednesday, February 05, 2025 at 04:34:53 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=\sin \left (x \right )^{2} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 44

dsolve(diff(y(x),x$2)+diff(y(x),x)+y(x)=sin(x)^2,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_2 +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_1 -\frac {\sin \left (2 x \right )}{13}+\frac {3 \cos \left (2 x \right )}{26}+\frac {1}{2} \]

✓ Solution by Mathematica

Time used: 0.566 (sec). Leaf size: 67

DSolve[D[y[x],{x,2}]+D[y[x],x]+y[x]==Sin[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {1}{13} \sin (2 x)+\frac {3}{26} \cos (2 x)+c_2 e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+c_1 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+\frac {1}{2} \]