23.4.236 problem 236
Internal
problem
ID
[6538]
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
:
236
Date
solved
:
Tuesday, September 30, 2025 at 03:02:50 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{2} y^{\prime \prime }&=a \end{align*}
✓ Maple. Time used: 0.032 (sec). Leaf size: 257
ode:=y(x)^2*diff(diff(y(x),x),x) = a;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {c_1 \left ({\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{2}-2 \textit {\_Z} \,c_1^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )} c_1^{2} a^{2}+{\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{2}-2 \textit {\_Z} \,c_1^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )}+2 a c_1 \right )}{2} \\
y &= \frac {c_1 \left ({\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{2}-2 \textit {\_Z} \,c_1^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )} c_1^{2} a^{2}+{\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{2}-2 \textit {\_Z} \,c_1^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )}+2 a c_1 \right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.097 (sec). Leaf size: 65
ode=y[x]^2*D[y[x],{x,2}] == a;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left (\frac {2 a \text {arctanh}\left (\frac {\sqrt {-\frac {2 a}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}+\frac {y(x) \sqrt {-\frac {2 a}{y(x)}+c_1}}{c_1}\right ){}^2=(x+c_2){}^2,y(x)\right ]
\]
✓ Sympy. Time used: 1.275 (sec). Leaf size: 357
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a + y(x)**2*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} \sqrt {2} C_{1} \sqrt {a} \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a} - 1} \sqrt {y{\left (x \right )}} + \frac {2 a \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 \sqrt {a}} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} y{\left (x \right )}}{a}}\right | > 2 \\\frac {\sqrt {2} i C_{1} \sqrt {a} \sqrt {y{\left (x \right )}}}{\sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a} + 1}} - \frac {\sqrt {2} i y^{\frac {3}{2}}{\left (x \right )}}{2 \sqrt {a} \sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a} + 1}} - \frac {2 i a \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 \sqrt {a}} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} + x, \ \begin {cases} \sqrt {2} C_{1} \sqrt {a} \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a} - 1} \sqrt {y{\left (x \right )}} + \frac {2 a \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 \sqrt {a}} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} y{\left (x \right )}}{a}}\right | > 2 \\\frac {\sqrt {2} i C_{1} \sqrt {a} \sqrt {y{\left (x \right )}}}{\sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a} + 1}} - \frac {\sqrt {2} i y^{\frac {3}{2}}{\left (x \right )}}{2 \sqrt {a} \sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a} + 1}} - \frac {2 i a \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 \sqrt {a}} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} - x\right ]
\]