29.6.25 problem 25

Internal problem ID [7310]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 6. SECOND-ORDER LINEAR EQUATIONSWITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO. page 422
Problem number : 25
Date solved : Tuesday, September 30, 2025 at 04:28:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y&=16 x^{2} {\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 34

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)-3*y(x) = 16*x^2*exp(-x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (6 c_2 \,{\mathrm e}^{4 x}-8 x^{3}-6 x^{2}+6 c_1 -3 x \right ) {\mathrm e}^{-x}}{6} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 50

ode=D[y[x],{x,2}]-2*D[y[x],x]-3*y[x]==16*x*Exp[-x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x} \left (e^{4 x} \int _1^x4 e^{-4 K[1]} K[1]dK[1]-2 x^2+c_2 e^{4 x}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.165 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-16*x**2*exp(-x) - 3*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} e^{3 x} + \left (C_{1} - \frac {4 x^{3}}{3} - x^{2} - \frac {x}{2}\right ) e^{- x} \]