29.20.13 problem 558
Internal
problem
ID
[5154]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
20
Problem
number
:
558
Date
solved
:
Sunday, March 30, 2025 at 06:45:40 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _Bernoulli]
\begin{align*} a x y y^{\prime }&=x^{2}+y^{2} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 64
ode:=a*x*y(x)*diff(y(x),x) = x^2+y(x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\sqrt {\left (a -1\right ) \left (c_1 \left (a -1\right ) x^{\frac {2}{a}}+x^{2}\right )}}{a -1} \\
y &= -\frac {\sqrt {\left (a -1\right ) \left (c_1 \left (a -1\right ) x^{\frac {2}{a}}+x^{2}\right )}}{a -1} \\
\end{align*}
✓ Mathematica. Time used: 4.539 (sec). Leaf size: 68
ode=a x y[x] D[y[x],x]==x^2+y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {x^2+(a-1) c_1 x^{2/a}}}{\sqrt {a-1}} \\
y(x)\to \frac {\sqrt {x^2+(a-1) c_1 x^{2/a}}}{\sqrt {a-1}} \\
\end{align*}
✓ Sympy. Time used: 2.236 (sec). Leaf size: 146
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*x*y(x)*Derivative(y(x), x) - x**2 - y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} - \sqrt {\frac {C_{1} a e^{\frac {2 \log {\left (x \right )}}{a}}}{a - 1} - \frac {C_{1} e^{\frac {2 \log {\left (x \right )}}{a}}}{a - 1} + \frac {x^{2}}{a - 1}} & \text {for}\: a > 1 \vee a < 1 \\- \sqrt {C_{1} e^{\frac {2 \log {\left (x \right )}}{a}} + \frac {2 e^{\frac {2 \log {\left (x \right )}}{a}} \log {\left (x \right )}}{a}} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \sqrt {\frac {C_{1} a e^{\frac {2 \log {\left (x \right )}}{a}}}{a - 1} - \frac {C_{1} e^{\frac {2 \log {\left (x \right )}}{a}}}{a - 1} + \frac {x^{2}}{a - 1}} & \text {for}\: a > 1 \vee a < 1 \\\sqrt {C_{1} e^{\frac {2 \log {\left (x \right )}}{a}} + \frac {2 e^{\frac {2 \log {\left (x \right )}}{a}} \log {\left (x \right )}}{a}} & \text {otherwise} \end {cases}\right ]
\]