56.1.43 problem 43

Internal problem ID [8755]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 43
Date solved : Monday, January 27, 2025 at 04:47:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+4 y&=\sin \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 39

dsolve(diff(y(x),x$2)+diff(y(x),x)+4*y(x)=sin(x),y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {15}\, x}{2}\right ) c_{2} +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {15}\, x}{2}\right ) c_{1} +\frac {3 \sin \left (x \right )}{10}-\frac {\cos \left (x \right )}{10} \]

✓ Solution by Mathematica

Time used: 1.571 (sec). Leaf size: 60

DSolve[D[y[x],{x,2}]+D[y[x],x]+4*y[x]==Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {3 \sin (x)}{10}-\frac {\cos (x)}{10}+c_2 e^{-x/2} \cos \left (\frac {\sqrt {15} x}{2}\right )+c_1 e^{-x/2} \sin \left (\frac {\sqrt {15} x}{2}\right ) \]