26.9.17 problem Exercise 22, problem 17, page 240

Internal problem ID [7126]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 22. Variation of Parameters
Problem number : Exercise 22, problem 17, page 240
Date solved : Tuesday, September 30, 2025 at 04:21:46 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\frac {2 y}{x^{2}}&=x \ln \left (x \right ) \end{align*}

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

ode:=diff(diff(y(x),x),x)-2/x*diff(y(x),x)+2/x^2*y(x) = x*ln(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{3} \ln \left (x \right )}{2}-\frac {3 x^{3}}{4}+c_2 \,x^{2}+c_1 x \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to \frac {1}{4} x \left (-3 x^2+2 x^2 \log (x)+4 c_2 x+4 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.178 (sec). Leaf size: 24

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

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