86.11.5 problem 5

Internal problem ID [23199]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 9. The operational method. Exercise 9c at page 137
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:24:17 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=8 \sin \left (2 x \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 8*sin(2*x); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-2 x}+2 \,{\mathrm e}^{-2 x} x -\cos \left (2 x \right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 23

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

\begin{align*} y(x)&\to e^{-2 x} (2 x+1)-\cos (2 x) \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = \left (2 x + 1\right ) e^{- 2 x} - \cos {\left (2 x \right )} \]