50.14.12 problem 2(d)

Internal problem ID [8050]
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)
Date solved : Monday, January 27, 2025 at 03:38:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+4 y&=\sin \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=1\\ y^{\prime }\left (\frac {\pi }{2}\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.306 (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)
 

\[ y = \frac {\left (\left (\sqrt {7}\, \sin \left (\frac {\sqrt {7}\, x}{2}\right )+35 \cos \left (\frac {\sqrt {7}\, x}{2}\right )\right ) \cos \left (\frac {\sqrt {7}\, \pi }{4}\right )-\sin \left (\frac {\sqrt {7}\, \pi }{4}\right ) \left (\sqrt {7}\, \cos \left (\frac {\sqrt {7}\, x}{2}\right )-35 \sin \left (\frac {\sqrt {7}\, x}{2}\right )\right )\right ) {\mathrm e}^{-\frac {3 x}{2}+\frac {3 \pi }{4}}}{42}-\frac {\cos \left (x \right )}{6}+\frac {\sin \left (x \right )}{6} \]

✓ Solution by Mathematica

Time used: 1.217 (sec). Leaf size: 79

DSolve[{D[y[x],{x,2}]+3*D[y[x],x]+4*y[x]==Sin[x],{y[Pi/2]==1,Derivative[1][y][Pi/2]==-1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{42} \left (-\sqrt {7} e^{\frac {3}{4} (\pi -2 x)} \sin \left (\frac {1}{4} \sqrt {7} (\pi -2 x)\right )+7 \sin (x)+35 e^{\frac {3}{4} (\pi -2 x)} \cos \left (\frac {1}{4} \sqrt {7} (\pi -2 x)\right )-7 \cos (x)\right ) \]