29.32.25 problem 960
Internal
problem
ID
[5537]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
32
Problem
number
:
960
Date
solved
:
Tuesday, March 04, 2025 at 09:51:35 PM
CAS
classification
:
[_quadrature]
\begin{align*} \left (x^{2}-a y\right ) {y^{\prime }}^{2}-2 x y y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.028 (sec). Leaf size: 28
ode:=(x^2-a*y(x))*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= -\frac {x^{2}}{a \operatorname {LambertW}\left (-\frac {c_{1} x^{2}}{a}\right )} \\
y \left (x \right ) &= c_{1} \\
\end{align*}
✓ Mathematica. Time used: 8.05 (sec). Leaf size: 310
ode=(x^2-a y[x]) (D[y[x],x])^2-2 x y[x] D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to c_1 \\
\text {Solve}\left [\frac {\left (2-\frac {2 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}\right ) \left (\frac {\frac {6 x^3}{x^2-a y(x)}-4 x}{\sqrt [3]{x^3}}+4\right ) \left (\left (1-\frac {x \left (2 a y(x)+x^2\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}\right ) \log \left (\frac {2-\frac {2 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}}{\sqrt [3]{2}}\right )+\left (\frac {2 a x y(x)+x^3}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}-1\right ) \log \left (\frac {\frac {\frac {6 x^3}{x^2-a y(x)}-4 x}{\sqrt [3]{x^3}}+4}{\sqrt [3]{2}}\right )-3\right )}{18 \sqrt [3]{2} \left (-\frac {\left (2 a y(x)+x^2\right )^3}{\left (x^2-a y(x)\right )^3}+\frac {3 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}-2\right )}&=\frac {2\ 2^{2/3} x \log (x)}{9 \sqrt [3]{x^3}}+c_1,y(x)\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 1.124 (sec). Leaf size: 26
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-2*x*y(x)*Derivative(y(x), x) + (-a*y(x) + x**2)*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = e^{\frac {- C_{1} + a W\left (- \frac {x^{2} e^{\frac {C_{1}}{a}}}{a}\right )}{a}}, \ y{\left (x \right )} = C_{1}\right ]
\]