89.19.14 problem 14

Internal problem ID [24742]
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 : 14
Date solved : Thursday, October 02, 2025 at 10:47:35 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=4-8 x +6 x \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+4*y(x) = 4-8*x+6*x*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x^{3}+c_1 x +c_2 \right ) {\mathrm e}^{2 x}-2 x -1 \]

✓ Mathematica. Time used: 0.251 (sec). Leaf size: 36

ode=D[y[x],{x,2}]-4*D[y[x],{x,1}]+4*y[x]==4-8*x+6*x*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.162 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-6*x*exp(2*x) + 8*x + 4*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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