29.33.10 problem 972
Internal
problem
ID
[5549]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
33
Problem
number
:
972
Date
solved
:
Tuesday, March 04, 2025 at 09:59:37 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational]
\begin{align*} y^{2} {y^{\prime }}^{2}-3 x y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.164 (sec). Leaf size: 117
ode:=y(x)^2*diff(y(x),x)^2-3*x*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {18^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}}}{2} \\
y \left (x \right ) &= -\frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (3 i 3^{{1}/{6}}+3^{{2}/{3}}\right )}{4} \\
y \left (x \right ) &= \frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (-3^{{2}/{3}}+3 i 3^{{1}/{6}}\right )}{4} \\
y \left (x \right ) &= 0 \\
y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+3 \sqrt {-4 \textit {\_a}^{3}+9}-9}{\textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} \right )+2 c_{1} \right ) x^{{2}/{3}} \\
\end{align*}
✓ Mathematica. Time used: 0.608 (sec). Leaf size: 208
ode=y[x]^2 (D[y[x],x])^2-3 x D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {3}{2} \log (y(x))-\frac {2 \sqrt {1-\frac {9 x^2}{4 y(x)^3}} \text {arcsinh}\left (\frac {3}{2} x \sqrt {-\frac {1}{y(x)^3}}\right )}{\sqrt {-\frac {1}{y(x)^3}} \sqrt {9 x^2-4 y(x)^3}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {2 \sqrt {1-\frac {9 x^2}{4 y(x)^3}} \text {arcsinh}\left (\frac {3}{2} x \sqrt {-\frac {1}{y(x)^3}}\right )}{\sqrt {-\frac {1}{y(x)^3}} \sqrt {9 x^2-4 y(x)^3}}+\frac {3}{2} \log (y(x))&=c_1,y(x)\right ] \\
y(x)\to \left (-\frac {3}{2}\right )^{2/3} x^{2/3} \\
y(x)\to \left (\frac {3}{2}\right )^{2/3} x^{2/3} \\
y(x)\to -\sqrt [3]{-1} \left (\frac {3}{2}\right )^{2/3} x^{2/3} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-3*x*Derivative(y(x), x) + y(x)**2*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(3*x + sqrt(9*x**2 - 4*y(x)**3))/(2*y(x)**2) + Derivative(y(x), x) cannot be solved by the factorable group method