29.10.8 problem 274
Internal
problem
ID
[4874]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
10
Problem
number
:
274
Date
solved
:
Sunday, March 30, 2025 at 04:08:17 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational, _Bernoulli]
\begin{align*} x^{2} y^{\prime }&=\left (a x +b y^{3}\right ) y \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 172
ode:=x^2*diff(y(x),x) = (a*x+b*y(x)^3)*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {27^{{1}/{3}} {\left (\left (c_1 \left (a -\frac {1}{3}\right ) x^{-3 a +1}-b \right )^{2} \left (a -\frac {1}{3}\right ) x \right )}^{{1}/{3}}}{c_1 \left (3 a -1\right ) x^{-3 a +1}-3 b} \\
y &= -\frac {27^{{1}/{3}} {\left (\left (c_1 \left (a -\frac {1}{3}\right ) x^{-3 a +1}-b \right )^{2} \left (a -\frac {1}{3}\right ) x \right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{\left (6 a -2\right ) c_1 \,x^{-3 a +1}-6 b} \\
y &= \frac {27^{{1}/{3}} {\left (\left (c_1 \left (a -\frac {1}{3}\right ) x^{-3 a +1}-b \right )^{2} \left (a -\frac {1}{3}\right ) x \right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{\left (6 a -2\right ) c_1 \,x^{-3 a +1}-6 b} \\
\end{align*}
✓ Mathematica. Time used: 3.536 (sec). Leaf size: 149
ode=x^2 D[y[x],x]==(a x+b y[x]^3)y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{(1-3 a) x^{3 a+1}}}{\sqrt [3]{3 b x^{3 a}+(1-3 a) c_1 x}} \\
y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{(1-3 a) x^{3 a+1}}}{\sqrt [3]{3 b x^{3 a}+(1-3 a) c_1 x}} \\
y(x)\to \frac {(-1)^{2/3} \sqrt [3]{(1-3 a) x^{3 a+1}}}{\sqrt [3]{3 b x^{3 a}+(1-3 a) c_1 x}} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 32.454 (sec). Leaf size: 139
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), x) - (a*x + b*y(x)**3)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{\frac {e^{3 a \log {\left (x \right )}}}{C_{1} - 3 b \left (\begin {cases} \frac {e^{3 a \log {\left (x \right )}}}{3 a x - x} & \text {for}\: a \neq \frac {1}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right )}}, \ y{\left (x \right )} = \frac {\sqrt [3]{\frac {e^{3 a \log {\left (x \right )}}}{C_{1} - 3 b \left (\begin {cases} \frac {e^{3 a \log {\left (x \right )}}}{3 a x - x} & \text {for}\: a \neq \frac {1}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right )}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{\frac {e^{3 a \log {\left (x \right )}}}{C_{1} - 3 b \left (\begin {cases} \frac {e^{3 a \log {\left (x \right )}}}{3 a x - x} & \text {for}\: a \neq \frac {1}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right )}} \left (-1 + \sqrt {3} i\right )}{2}\right ]
\]