29.22.1 problem 607
Internal
problem
ID
[5201]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
22
Problem
number
:
607
Date
solved
:
Sunday, March 30, 2025 at 06:51:50 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (3 x^{2}-y^{2}\right ) y^{\prime }&=2 x y \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 313
ode:=(3*x^2-y(x)^2)*diff(y(x),x) = 2*x*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {1+\frac {\left (-108 x^{2} c_1^{2}+12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 +8\right )^{{1}/{3}}}{2}+\frac {2}{\left (-108 x^{2} c_1^{2}+12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 +8\right )^{{1}/{3}}}}{3 c_1} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (-108 x^{2} c_1^{2}+12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 +8\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (-108 x^{2} c_1^{2}+12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 +8\right )^{{1}/{3}}+4}{12 \left (-108 x^{2} c_1^{2}+12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 +8\right )^{{1}/{3}} c_1} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-108 x^{2} c_1^{2}+12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 +8\right )^{{2}/{3}}-4 i \sqrt {3}+4 \left (-108 x^{2} c_1^{2}+12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 +8\right )^{{1}/{3}}-4}{12 \left (-108 x^{2} c_1^{2}+12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 +8\right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 60.158 (sec). Leaf size: 458
ode=(3 x^2-y[x]^2)D[y[x],x]==2 x y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{3} \left (\frac {\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-e^{c_1}\right ) \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\
y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x*y(x) + (3*x**2 - y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out