29.32.11 problem 945
Internal
problem
ID
[5523]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
32
Problem
number
:
945
Date
solved
:
Sunday, March 30, 2025 at 08:29:29 AM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} y {y^{\prime }}^{2}-4 a^{2} x y^{\prime }+a^{2} y&=0 \end{align*}
✓ Maple. Time used: 0.196 (sec). Leaf size: 122
ode:=y(x)*diff(y(x),x)^2-4*a^2*x*diff(y(x),x)+a^2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}-2 a^{2}+\sqrt {-\textit {\_a}^{2} a^{2}+4 a^{4}}}{\textit {\_a} \left (\textit {\_a}^{2}-3 a^{2}\right )}d \textit {\_a} +c_1 \right ) x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2}-2 a^{2}-\sqrt {-\textit {\_a}^{2} a^{2}+4 a^{4}}}{\textit {\_a} \left (\textit {\_a}^{2}-3 a^{2}\right )}d \textit {\_a} +c_1 \right ) x \\
\end{align*}
✓ Mathematica. Time used: 8.798 (sec). Leaf size: 758
ode=y[x] (D[y[x],x])^2-4 a^2 x D[y[x],x]+a^2 y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {8 \left (4 a^2-\frac {y(x)^2}{x^2}\right )^{3/2} \text {arcsinh}\left (\frac {\sqrt {\frac {y(x)}{x}-2 a}}{2 \sqrt {a}}\right )+\sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \left (\sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )-8 \arctan \left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )+4 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \text {arctanh}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )-2 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \text {arctanh}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )\right )}{6 \sqrt {a} \sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {\frac {y(x)}{a x}+2} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}&=-\log (x)+c_1,y(x)\right ] \\
\text {Solve}\left [\frac {\sqrt {a} \sqrt {\frac {y(x)}{a x}+2} \left (\sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {4 a^2-\frac {y(x)^2}{x^2}} \left (\log \left (3 a^2-\frac {y(x)^2}{x^2}\right )+8 \arctan \left (\frac {\sqrt {2 a-\frac {y(x)}{x}}}{\sqrt {2 a+\frac {y(x)}{x}}}\right )+4 \log \left (\frac {y(x)}{x}\right )\right )-4 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \text {arctanh}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{2 a}\right )+2 \sqrt {\frac {y(x)}{x}-2 a} \left (\frac {y(x)^2}{x^2}-4 a^2\right ) \text {arctanh}\left (\frac {\sqrt {4 a^2-\frac {y(x)^2}{x^2}}}{a}\right )\right )-8 \left (4 a^2-\frac {y(x)^2}{x^2}\right )^{3/2} \text {arcsinh}\left (\frac {\sqrt {\frac {y(x)}{x}-2 a}}{2 \sqrt {a}}\right )}{6 \sqrt {a} \sqrt {-\left (\frac {y(x)}{x}-2 a\right )^2} \sqrt {2 a+\frac {y(x)}{x}} \sqrt {\frac {y(x)}{a x}+2} \sqrt {4 a^2-\frac {y(x)^2}{x^2}}}&=-\log (x)+c_1,y(x)\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 13.151 (sec). Leaf size: 129
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-4*a**2*x*Derivative(y(x), x) + a**2*y(x) + y(x)*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {3} \sqrt {a^{2} x^{2}}, \ y{\left (x \right )} = \sqrt {3} \sqrt {a^{2} x^{2}}, \ \log {\left (x \right )} = C_{1} + \log {\left (\frac {\sqrt {2} \sqrt [6]{a - \sqrt {4 a^{2} - \frac {y^{2}{\left (x \right )}}{x^{2}}}}}{2 \sqrt [6]{2 a + \sqrt {4 a^{2} - \frac {y^{2}{\left (x \right )}}{x^{2}}}} \sqrt {2 a^{2} + a \sqrt {4 a^{2} - \frac {y^{2}{\left (x \right )}}{x^{2}}} - \frac {y^{2}{\left (x \right )}}{x^{2}}}} \right )}\right ]
\]