23.1.195 problem 193 (b)
Internal
problem
ID
[4802]
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
:
193
(b)
Date
solved
:
Tuesday, September 30, 2025 at 08:41:06 AM
CAS
classification
:
[_rational, _Bernoulli]
\begin{align*} x y^{\prime }&=a y+b \left (-x^{2}+1\right ) y^{3} \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 146
ode:=x*diff(y(x),x) = a*y(x)+b*(-x^2+1)*y(x)^3;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\sqrt {x^{2 a} a \left (a +1\right ) \left (a b \,x^{2+2 a}+\left (a +1\right ) \left (c_1 a -b \,x^{2 a}\right )\right )}}{a b \,x^{2+2 a}+\left (a +1\right ) \left (c_1 a -b \,x^{2 a}\right )} \\
y &= -\frac {\sqrt {x^{2 a} a \left (a +1\right ) \left (a b \,x^{2+2 a}+\left (a +1\right ) \left (c_1 a -b \,x^{2 a}\right )\right )}}{a b \,x^{2+2 a}+\left (a +1\right ) \left (c_1 a -b \,x^{2 a}\right )} \\
\end{align*}
✓ Mathematica. Time used: 4.56 (sec). Leaf size: 112
ode=x*D[y[x],x]==a*y[x]+b*(1-x^2)*y[x]^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {i \sqrt {a} \sqrt {a+1} x^a}{\sqrt {-b \left (a \left (x^2-1\right )-1\right ) x^{2 a}-a (a+1) c_1}}\\ y(x)&\to \frac {i \sqrt {a} \sqrt {a+1} x^a}{\sqrt {-b \left (a \left (x^2-1\right )-1\right ) x^{2 a}-a (a+1) c_1}}\\ y(x)&\to 0 \end{align*}
✓ Sympy. Time used: 2.978 (sec). Leaf size: 325
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a*y(x) - b*(1 - x**2)*y(x)**3 + x*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} - \sqrt {\frac {1}{C_{1} x^{2} + 2 b x^{2} \log {\left (x \right )} + b}} & \text {for}\: a = -1 \\- \sqrt {\frac {1}{C_{1} + b x^{2} - 2 b \log {\left (x \right )}}} & \text {for}\: a = 0 \\- \sqrt {\frac {a^{2} e^{2 a \log {\left (x \right )}}}{C_{1} a^{2} + C_{1} a + a b x^{2} e^{2 a \log {\left (x \right )}} - a b e^{2 a \log {\left (x \right )}} - b e^{2 a \log {\left (x \right )}}} + \frac {a e^{2 a \log {\left (x \right )}}}{C_{1} a^{2} + C_{1} a + a b x^{2} e^{2 a \log {\left (x \right )}} - a b e^{2 a \log {\left (x \right )}} - b e^{2 a \log {\left (x \right )}}}} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \sqrt {\frac {1}{C_{1} x^{2} + 2 b x^{2} \log {\left (x \right )} + b}} & \text {for}\: a = -1 \\\sqrt {\frac {1}{C_{1} + b x^{2} - 2 b \log {\left (x \right )}}} & \text {for}\: a = 0 \\\sqrt {\frac {a^{2} e^{2 a \log {\left (x \right )}}}{C_{1} a^{2} + C_{1} a + a b x^{2} e^{2 a \log {\left (x \right )}} - a b e^{2 a \log {\left (x \right )}} - b e^{2 a \log {\left (x \right )}}} + \frac {a e^{2 a \log {\left (x \right )}}}{C_{1} a^{2} + C_{1} a + a b x^{2} e^{2 a \log {\left (x \right )}} - a b e^{2 a \log {\left (x \right )}} - b e^{2 a \log {\left (x \right )}}}} & \text {otherwise} \end {cases}\right ]
\]