86.11.1 problem 1

Internal problem ID [23195]
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 : 1
Date solved : Thursday, October 02, 2025 at 09:24:14 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=5 \sin \left (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: 0.013 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+2*y(x) = 5*sin(x); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 29

ode=D[y[x],{x,2}]+2*D[y[x],x]+2*y[x]==5*Sin[x]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.135 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*y(x) - 5*sin(x) + 2*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)
 

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