23.3.485 problem 491

Internal problem ID [6199]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 491
Date solved : Tuesday, September 30, 2025 at 02:36:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y+3 x^{2} y^{\prime }+x^{3} y^{\prime \prime }&=1 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

ode:=x*y(x)+3*x^2*diff(y(x),x)+x^3*diff(diff(y(x),x),x) = 1; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 27

ode=x*y[x] + 3*x^2*D[y[x],x] + x^3*D[y[x],{x,2}] == 1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.164 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 2)) + 3*x**2*Derivative(y(x), x) + x*y(x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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