29.22.18 problem 626
Internal
problem
ID
[5218]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
22
Problem
number
:
626
Date
solved
:
Sunday, March 30, 2025 at 06:54:41 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (x^{2}-3 y^{2}\right ) y^{\prime }+1+2 x y&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 294
ode:=(x^2-3*y(x)^2)*diff(y(x),x)+1+2*x*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (108 c_1 +108 x +12 \sqrt {-12 x^{6}+81 c_1^{2}+162 c_1 x +81 x^{2}}\right )^{{2}/{3}}+12 x^{2}}{6 \left (108 c_1 +108 x +12 \sqrt {-12 x^{6}+81 c_1^{2}+162 c_1 x +81 x^{2}}\right )^{{1}/{3}}} \\
y &= \frac {-i \sqrt {3}\, \left (108 c_1 +108 x +12 \sqrt {-12 x^{6}+81 c_1^{2}+162 c_1 x +81 x^{2}}\right )^{{2}/{3}}+12 i \sqrt {3}\, x^{2}-\left (108 c_1 +108 x +12 \sqrt {-12 x^{6}+81 c_1^{2}+162 c_1 x +81 x^{2}}\right )^{{2}/{3}}-12 x^{2}}{12 \left (108 c_1 +108 x +12 \sqrt {-12 x^{6}+81 c_1^{2}+162 c_1 x +81 x^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\left (108 c_1 +108 x +12 \sqrt {-12 x^{6}+81 c_1^{2}+162 c_1 x +81 x^{2}}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{12}-\frac {\left (1+i \sqrt {3}\right ) x^{2}}{\left (108 c_1 +108 x +12 \sqrt {-12 x^{6}+81 c_1^{2}+162 c_1 x +81 x^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 4.796 (sec). Leaf size: 307
ode=(x^2-3 y[x]^2)D[y[x],x]+1+2 x y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*y(x) + (x**2 - 3*y(x)**2)*Derivative(y(x), x) + 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out