23.4.254 problem 254

Internal problem ID [6556]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 254
Date solved : Friday, October 03, 2025 at 02:09:29 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.206 (sec). Leaf size: 179

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

\begin{align*} y &= \left (18 a \,c_1^{2}+\operatorname {RootOf}\left (27 a \ln \left (\frac {9 a \,c_1^{2}+\sqrt {\frac {\textit {\_Z} \left (18 a \,c_1^{2}+\textit {\_Z} \right )}{c_1^{2}}}\, c_1 +\textit {\_Z}}{c_1}\right ) x \,c_1^{3}+3 \sqrt {\frac {\textit {\_Z} \left (18 a \,c_1^{2}+\textit {\_Z} \right )}{c_1^{2}}}\, \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \,c_1^{2}+3 c_2 x \,\operatorname {csgn}\left (\frac {1}{c_1}\right )+\operatorname {csgn}\left (\frac {1}{c_1}\right )\right )\right ) x \\ y &= \left (18 a \,c_1^{2}+\operatorname {RootOf}\left (27 a \ln \left (\frac {9 a \,c_1^{2}+\sqrt {\frac {\textit {\_Z} \left (18 a \,c_1^{2}+\textit {\_Z} \right )}{c_1^{2}}}\, c_1 +\textit {\_Z}}{c_1}\right ) x \,c_1^{3}+3 \sqrt {\frac {\textit {\_Z} \left (18 a \,c_1^{2}+\textit {\_Z} \right )}{c_1^{2}}}\, \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \,c_1^{2}-3 c_2 x \,\operatorname {csgn}\left (\frac {1}{c_1}\right )-\operatorname {csgn}\left (\frac {1}{c_1}\right )\right )\right ) x \\ \end{align*}

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 116

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

\[ \text {Solve}\left [-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c_1} \left (\frac {y(x)}{x}+\frac {a}{2 c_1}\right )}{\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}\right )}{2 \sqrt {2} c_1{}^{3/2}}-\frac {\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}{2 c_1}-\frac {1}{x}-c_2=0,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : solve: Cannot solve -a + x*y(x)**2*Derivative(y(x), (x, 2))