83.12.6 problem 6

Internal problem ID [22007]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter VII. Operational method. Ex. XV at page 121
Problem number : 6
Date solved : Thursday, October 02, 2025 at 08:21:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=2 \,{\mathrm e}^{-x} \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+9*y(x) = 2*exp(-x)*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 x +c_2 \right ) {\mathrm e}^{-3 x}-\frac {8 \left (\cos \left (x \right )-\frac {3 \sin \left (x \right )}{4}\right ) {\mathrm e}^{-x}}{25} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 42

ode=D[y[x],{x,2}]+6*D[y[x],x]+9*y[x]==2*Exp[-x]*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{25} e^{-3 x} \left (6 e^{2 x} \sin (x)-8 e^{2 x} \cos (x)+25 (c_2 x+c_1)\right ) \end{align*}

✓ Sympy. Time used: 0.197 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*y(x) + 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 2*exp(-x)*sin(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\left (C_{1} + C_{2} x\right ) e^{- 2 x} + \frac {6 \sin {\left (x \right )}}{25} - \frac {8 \cos {\left (x \right )}}{25}\right ) e^{- x} \]