87.18.21 problem 21

Internal problem ID [23662]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 135
Problem number : 21
Date solved : Thursday, October 02, 2025 at 09:44:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=\frac {{\mathrm e}^{-3 x}}{x^{3}} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=4 \,{\mathrm e}^{-3} \\ y^{\prime }\left (1\right )&=-2 \,{\mathrm e}^{-3} \\ \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+9*y(x) = 1/x^3*exp(-3*x); 
ic:=[y(1) = 4*exp(-3), D(y)(1) = -2*exp(-3)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-3 x} \left (-7+\frac {21 x}{2}+\frac {1}{2 x}\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 27

ode=D[y[x],{x,2}]+6*D[y[x],x]+9*y[x]==1/x^3*Exp[-3*x]; 
ic={y[1]==4*Exp[-3],Derivative[1][y][1] ==-2*Exp[-3]}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-3 x} \left (21 x^2-14 x+1\right )}{2 x} \end{align*}

✓ Sympy. Time used: 0.299 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*y(x) + 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(-3*x)/x**3,0) 
ics = {y(1): 4*exp(-3), Subs(Derivative(y(x), x), x, 1): -2*exp(-3)} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {21 x}{2} - 7 + \frac {1}{2 x}\right ) e^{- 3 x} \]