89.19.13 problem 13

Internal problem ID [24741]
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 : 13
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&=20-3 x \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+4*y(x) = 20-3*x*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = 5+\frac {\left (-x^{3}+2 c_1 x +2 c_2 \right ) {\mathrm e}^{2 x}}{2} \]

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 27

ode=D[y[x],{x,2}]-4*D[y[x],{x,1}]+4*y[x]==20-3*x*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.171 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x*exp(2*x) + 4*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 20,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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