83.12.1 problem 1

Internal problem ID [22002]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter VII. Operational method. Ex. XV at page 121
Problem number : 1
Date solved : Thursday, October 02, 2025 at 08:21:34 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

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

\[ y = {\mathrm e}^{2 x} \left (c_2 +c_1 x +\frac {1}{6} x^{3}\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 27

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

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

✓ Sympy. Time used: 0.156 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x^{2}}{6}\right )\right ) e^{2 x} \]