83.8.26 problem 27
Internal
problem
ID
[19081]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
II.
Equations
of
first
order
and
first
degree.
Misc
examples
on
chapter
II
at
page
25
Problem
number
:
27
Date
solved
:
Monday, March 31, 2025 at 06:47:41 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} x^{2}-a y&=\left (a x -y^{2}\right ) y^{\prime } \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 280
ode:=x^2-a*y(x) = (a*x-y(x)^2)*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {4 a x +\left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_1 \right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}}{2 \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_1 \right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\left (-i \sqrt {3}-1\right ) \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_1 \right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}}{4}+\frac {\left (i \sqrt {3}-1\right ) x a}{\left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_1 \right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_1 \right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}}{4}-\frac {\left (1+i \sqrt {3}\right ) x a}{\left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (-4 a^{3}+6 c_1 \right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 5.52 (sec). Leaf size: 325
ode=(x^2-a*y[x])==(a*x-y[x]^2)*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {2 a x+\sqrt [3]{2} \left (\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1}} \\
y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1\right ){}^{2/3}+2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) a x}{4 \sqrt [3]{\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1}} \\
y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1\right ){}^{2/3}+2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) a x}{4 \sqrt [3]{\sqrt {-4 a^3 x^3+\left (x^3+3 c_1\right ){}^2}+x^3+3 c_1}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*y(x) + x**2 - (a*x - y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out