84.27.3 problem 16.3

Internal problem ID [22276]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 16. Initial-value problems. Solved problems. Page 81
Problem number : 16.3
Date solved : Thursday, October 02, 2025 at 08:36:59 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+8 y&=\sin \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+8*y(x) = sin(x); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (138 \cos \left (x \right )^{2}+131 \cos \left (x \right ) \sin \left (x \right )-69\right ) {\mathrm e}^{-2 x}}{65}+\frac {7 \sin \left (x \right )}{65}-\frac {4 \cos \left (x \right )}{65} \]

✓ Mathematica. Time used: 0.135 (sec). Leaf size: 45

ode=D[y[x],{x,2}]+4*D[y[x],{x,1}]+8*y[x]==Sin[x]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{65} e^{-2 x} \left (7 e^{2 x} \sin (x)+69 \cos (2 x)+\left (131 \sin (x)-4 e^{2 x}\right ) \cos (x)\right ) \end{align*}

✓ Sympy. Time used: 0.157 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*y(x) - sin(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {131 \sin {\left (2 x \right )}}{130} + \frac {69 \cos {\left (2 x \right )}}{65}\right ) e^{- 2 x} + \frac {7 \sin {\left (x \right )}}{65} - \frac {4 \cos {\left (x \right )}}{65} \]