66.1.47 problem Problem 61
Internal
problem
ID
[13823]
Book
:
Differential
equations
and
the
calculus
of
variations
by
L.
ElSGOLTS.
MIR
PUBLISHERS,
MOSCOW,
Third
printing
1977.
Section
:
Chapter
1,
First-Order
Differential
Equations.
Problems
page
88
Problem
number
:
Problem
61
Date
solved
:
Monday, March 31, 2025 at 08:14:14 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (y^{2}-x \right ) y^{\prime }-y+x^{2}&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 316
ode:=(y(x)^2-x)*diff(y(x),x)-y(x)+x^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 x}{2 \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (-\left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 x \right ) \sqrt {3}-\left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x}{4 \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (\left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x \right ) \sqrt {3}-\left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x}{4 \left (-4 x^{3}-12 c_1 +4 \sqrt {x^{6}+\left (6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 3.707 (sec). Leaf size: 326
ode=(y[x]^2-x)*D[y[x],x]-y[x]+x^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {2 x+\sqrt [3]{2} \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\
y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x}{4 \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\
y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x}{4 \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2 + (-x + y(x)**2)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out