78.8.16 problem 16

Internal problem ID [18135]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 16
Date solved : Thursday, March 13, 2025 at 11:40:17 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }&=1 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 16

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x) = 1; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 21

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

\[ y(x)\to \frac {\log ^2(x)}{2}+c_1 \log (x)+c_2 \]

✓ Sympy. Time used: 0.177 (sec). Leaf size: 15

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

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