60.7.161 problem 1779 (book 6.188)
Internal
problem
ID
[11711]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1779
(book
6.188)
Date
solved
:
Sunday, March 30, 2025 at 08:43:14 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{2} y^{\prime \prime }-a&=0 \end{align*}
✓ Maple. Time used: 0.071 (sec). Leaf size: 257
ode:=y(x)^2*diff(diff(y(x),x),x)-a = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {c_1 \left (2 a c_1 +{\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 )}+{\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}\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 )}+2 a c_1 +{\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}\right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.175 (sec). Leaf size: 65
ode=-a + y[x]^2*D[y[x],{x,2}] == 0;
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: 2.083 (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)
\[
\left [ \begin {cases} \sqrt {2} C_{1} \sqrt {a} \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a} - 1} \sqrt {y{\left (x \right )}} + \frac {2 a \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 \sqrt {a}} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} y{\left (x \right )}}{a}}\right | > 2 \\\frac {\sqrt {2} i C_{1} \sqrt {a} \sqrt {y{\left (x \right )}}}{\sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a} + 1}} - \frac {\sqrt {2} i y^{\frac {3}{2}}{\left (x \right )}}{2 \sqrt {a} \sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a} + 1}} - \frac {2 i a \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 \sqrt {a}} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} + x, \ \begin {cases} \sqrt {2} C_{1} \sqrt {a} \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a} - 1} \sqrt {y{\left (x \right )}} + \frac {2 a \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 \sqrt {a}} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} y{\left (x \right )}}{a}}\right | > 2 \\\frac {\sqrt {2} i C_{1} \sqrt {a} \sqrt {y{\left (x \right )}}}{\sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a} + 1}} - \frac {\sqrt {2} i y^{\frac {3}{2}}{\left (x \right )}}{2 \sqrt {a} \sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a} + 1}} - \frac {2 i a \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 \sqrt {a}} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} - x\right ]
\]