44.14.12 problem 2(d)

Internal problem ID [9336]
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 : Tuesday, September 30, 2025 at 06:16:33 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*}

✓ Maple. Time used: 0.197 (sec). Leaf size: 83

ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)+4*y(x) = sin(x); 
ic:=[y(1/2*Pi) = 1, D(y)(1/2*Pi) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.265 (sec). Leaf size: 277

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

\begin{align*} y(x)&\to -\frac {1}{7} e^{-3 x/2} \left (7 \sin \left (\frac {\sqrt {7} x}{2}\right ) \int _1^{\frac {\pi }{2}}\frac {2 e^{\frac {3 K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {7} K[1]\right ) \sin (K[1])}{\sqrt {7}}dK[1]-7 \sin \left (\frac {\sqrt {7} x}{2}\right ) \int _1^x\frac {2 e^{\frac {3 K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {7} K[1]\right ) \sin (K[1])}{\sqrt {7}}dK[1]+7 \cos \left (\frac {\sqrt {7} x}{2}\right ) \int _1^{\frac {\pi }{2}}-\frac {2 e^{\frac {3 K[2]}{2}} \sin (K[2]) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )}{\sqrt {7}}dK[2]-7 \cos \left (\frac {\sqrt {7} x}{2}\right ) \int _1^x-\frac {2 e^{\frac {3 K[2]}{2}} \sin (K[2]) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )}{\sqrt {7}}dK[2]+\sqrt {7} e^{3 \pi /4} \sin \left (\frac {1}{4} \sqrt {7} (\pi -2 x)\right )-7 e^{3 \pi /4} \cos \left (\frac {1}{4} \sqrt {7} (\pi -2 x)\right )\right ) \end{align*}

✓ Sympy. Time used: 0.225 (sec). Leaf size: 230

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - sin(x) + 3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(pi/2): 1, Subs(Derivative(y(x), x), x, pi/2): -1} 
dsolve(ode,func=y(x),ics=ics)
 

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