88.18.9 problem 8 (b)

Internal problem ID [24128]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 133
Problem number : 8 (b)
Date solved : Thursday, October 02, 2025 at 10:00:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \ln \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 25

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

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

✓ Sympy. Time used: 0.146 (sec). Leaf size: 17

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

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