87.18.23 problem 23

Internal problem ID [23664]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 135
Problem number : 23
Date solved : Thursday, October 02, 2025 at 09:44:02 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ y^{\prime }\left (1\right )&={\mathrm e}^{2} \\ \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+4*y(x) = exp(2*x)/x^4; 
ic:=[y(1) = 0, D(y)(1) = exp(2)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{2 x} \left (-\frac {3}{2}+\frac {4 x}{3}+\frac {1}{6 x^{2}}\right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 29

ode=D[y[x],{x,2}]-4*D[y[x],x]+4*y[x]==1/x^4*Exp[2*x]; 
ic={y[1]==0,Derivative[1][y][1] ==Exp[2]}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{2 x} \left (8 x^3-9 x^2+1\right )}{6 x^2} \end{align*}

✓ Sympy. Time used: 0.206 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(2*x)/x**4,0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): exp(2)} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {4 x}{3} - \frac {3}{2} + \frac {1}{6 x^{2}}\right ) e^{2 x} \]