81.2.9 problem 3-10
Internal
problem
ID
[21498]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
I.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
3.
Exact
differential
equations.
Page
42.
Problem
number
:
3-10
Date
solved
:
Thursday, October 02, 2025 at 07:42:10 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (x +y^{2}\right ) y^{\prime }+y-x^{2}&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 318
ode:=(x+y(x)^2)*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: 3.117 (sec). Leaf size: 329
ode=(x+y[x]^2)*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 \sqrt [3]{2} x+2^{2/3} \left (x^3+\sqrt {x^6+(4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2 \sqrt [3]{x^3+\sqrt {x^6+(4+6 c_1) x^3+9 c_1{}^2}+3 c_1}}\\ y(x)&\to \frac {i 2^{2/3} \left (\sqrt {3}+i\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