89.19.19 problem 19

Internal problem ID [24747]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 151
Problem number : 19
Date solved : Thursday, October 02, 2025 at 10:47:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y&=8 x \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x)-4*y(x) = 8*x*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 34

ode=D[y[x],{x,2}]-4*y[x]==8*x*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.095 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x*exp(2*x) - 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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