29.34.12 problem 1014
Internal
problem
ID
[5583]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1014
Date
solved
:
Sunday, March 30, 2025 at 09:04:43 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} 9 \left (-x^{2}+1\right ) y^{4} {y^{\prime }}^{2}+6 x y^{5} y^{\prime }+4 x^{2}&=0 \end{align*}
✓ Maple. Time used: 0.541 (sec). Leaf size: 225
ode:=9*(-x^2+1)*y(x)^4*diff(y(x),x)^2+6*x*y(x)^5*diff(y(x),x)+4*x^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}} \\
y &= -2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\
y &= \frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\
y &= -\frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\
y &= \frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\
y &= \frac {2^{{2}/{3}} {\left (\left (-4 c_1^{2}+x^{2}-1\right ) c_1^{2}\right )}^{{1}/{3}}}{2 c_1} \\
y &= -\frac {2^{{2}/{3}} {\left (\left (-4 c_1^{2}+x^{2}-1\right ) c_1^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 c_1} \\
y &= \frac {2^{{2}/{3}} {\left (\left (-4 c_1^{2}+x^{2}-1\right ) c_1^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 c_1} \\
\end{align*}
✓ Mathematica. Time used: 0.432 (sec). Leaf size: 199
ode=9(1-x^2) y[x]^4 (D[y[x],x])^2 +6 x y[x]^5 D[y[x],x]+4 x^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-4 x^2+4+c_1{}^2}}{\sqrt [3]{c_1}} \\
y(x)\to -1 \\
y(x)\to 0 \\
y(x)\to \sqrt [3]{-\frac {1}{2}} \\
y(x)\to \text {Indeterminate} \\
y(x)\to -\sqrt [3]{-2} \sqrt [6]{1-x^2} \\
y(x)\to \sqrt [3]{-2} \sqrt [6]{1-x^2} \\
y(x)\to -\sqrt [3]{2} \sqrt [6]{1-x^2} \\
y(x)\to \sqrt [3]{2} \sqrt [6]{1-x^2} \\
y(x)\to -(-1)^{2/3} \sqrt [3]{2} \sqrt [6]{1-x^2} \\
y(x)\to (-1)^{2/3} \sqrt [3]{2} \sqrt [6]{1-x^2} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*x**2 + 6*x*y(x)**5*Derivative(y(x), x) + (9 - 9*x**2)*y(x)**4*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out