23.1.715 problem 711
Internal
problem
ID
[5322]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
711
Date
solved
:
Tuesday, September 30, 2025 at 12:29:40 PM
CAS
classification
:
[_rational]
\begin{align*} \left (a^{2} x^{2}+\left (x^{2}+y^{2}\right )^{2}\right ) y^{\prime }&=a^{2} x y \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 197
ode:=(a^2*x^2+(x^2+y(x)^2)^2)*diff(y(x),x) = a^2*x*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {-2 a^{2}-2 x^{2}-2 \sqrt {x^{4}+\left (2 a^{2}-2 c_1 \right ) x^{2}+\left (a^{2}+c_1 \right )^{2}}-2 c_1}}{2} \\
y &= \frac {\sqrt {-2 a^{2}-2 x^{2}-2 \sqrt {x^{4}+\left (2 a^{2}-2 c_1 \right ) x^{2}+\left (a^{2}+c_1 \right )^{2}}-2 c_1}}{2} \\
y &= -\frac {\sqrt {-2 a^{2}-2 x^{2}+2 \sqrt {x^{4}+\left (2 a^{2}-2 c_1 \right ) x^{2}+\left (a^{2}+c_1 \right )^{2}}-2 c_1}}{2} \\
y &= \frac {\sqrt {-2 a^{2}-2 x^{2}+2 \sqrt {x^{4}+\left (2 a^{2}-2 c_1 \right ) x^{2}+\left (a^{2}+c_1 \right )^{2}}-2 c_1}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.18 (sec). Leaf size: 381
ode=(a^2 x^2+(x^2+y[x]^2)^2)D[y[x],x]==a^2 x y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (\exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 \left (a^2+K[1]\right )}-\frac {3}{2 K[1]}\right )dK[1]\right ) x^4+a^2 \exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 \left (a^2+K[1]\right )}-\frac {3}{2 K[1]}\right )dK[1]\right ) x^2+2 \exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 \left (a^2+K[1]\right )}-\frac {3}{2 K[1]}\right )dK[1]\right ) K[3]^2 x^2+\exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 \left (a^2+K[1]\right )}-\frac {3}{2 K[1]}\right )dK[1]\right ) K[3]^4-\int _1^x\left (-\exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 \left (a^2+K[1]\right )}-\frac {3}{2 K[1]}\right )dK[1]\right ) K[2] a^2-2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 \left (a^2+K[1]\right )}-\frac {3}{2 K[1]}\right )dK[1]\right ) K[2] K[3]^2 \left (-\frac {1}{2 \left (a^2+K[2]^2+K[3]^2\right )}-\frac {3}{2 \left (K[2]^2+K[3]^2\right )}\right ) a^2\right )dK[2]\right )dK[3]+\int _1^x-a^2 \exp \left (\int _1^{K[2]^2+y(x)^2}\left (-\frac {1}{2 \left (a^2+K[1]\right )}-\frac {3}{2 K[1]}\right )dK[1]\right ) K[2] y(x)dK[2]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*x*y(x) + (a**2*x**2 + (x**2 + y(x)**2)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out