Internal problem ID [5629]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Problems for Review and Discovery. Drill excercises. Page 105
Problem number: 2(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+3 y^{\prime }+4 y-\sin \relax (x )=0} \end {gather*} With initial conditions \begin {align*} \left [y \left (\frac {\pi }{2}\right ) = 1, y^{\prime }\left (\frac {\pi }{2}\right ) = -1\right ] \end {align*}
✓ Solution by Maple
Time used: 0.266 (sec). Leaf size: 83
dsolve([diff(y(x),x$2)+3*diff(y(x),x)+4*y(x)=sin(x),y(1/2*Pi) = 1, D(y)(1/2*Pi) = -1],y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\left (\sqrt {7}\, \sin \left (\frac {\sqrt {7}\, x}{2}\right )+35 \cos \left (\frac {\sqrt {7}\, x}{2}\right )\right ) \cos \left (\frac {\sqrt {7}\, \pi }{4}\right )-\sin \left (\frac {\sqrt {7}\, \pi }{4}\right ) \left (\sqrt {7}\, \cos \left (\frac {\sqrt {7}\, x}{2}\right )-35 \sin \left (\frac {\sqrt {7}\, x}{2}\right )\right )\right ) {\mathrm e}^{-\frac {3 x}{2}+\frac {3 \pi }{4}}}{42}-\frac {\cos \relax (x )}{6}+\frac {\sin \relax (x )}{6} \]
✓ Solution by Mathematica
Time used: 0.764 (sec). Leaf size: 70
DSolve[{y''[x]+3*y'[x]+4*y[x]==Sin[x],{y[Pi/2]==1,y'[Pi/2]==-1}},y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{42} \left (7 \sin (x)-7 \cos (x)+e^{\frac {3}{4} (\pi -2 x)} \left (35 \cos \left (\frac {1}{4} \sqrt {7} (\pi -2 x)\right )-\sqrt {7} \sin \left (\frac {1}{4} \sqrt {7} (\pi -2 x)\right )\right )\right ) \\ \end{align*}