88.23.14 problem 16

Internal problem ID [24188]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Miscellaneous Exercises at page 162
Problem number : 16
Date solved : Thursday, October 02, 2025 at 10:00:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }-5 y&=x^{2} {\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)-5*y(x) = x^2*exp(-x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (x^{3}+\frac {x^{2}}{2}-18 c_1 \,{\mathrm e}^{6 x}+\frac {x}{6}-18 c_2 \right ) {\mathrm e}^{-x}}{18} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 42

ode=D[y[x],{x,2}]-4*D[y[x],x]-5*y[x]==x^2*Exp[-x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 e^{5 x}-\frac {1}{648} e^{-x} \left (36 x^3+18 x^2+6 x+1-648 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.174 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*exp(-x) - 5*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} e^{5 x} + \left (C_{1} - \frac {x^{3}}{18} - \frac {x^{2}}{36} - \frac {x}{108}\right ) e^{- x} \]