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
:
Monday, January 27, 2025 at 12:12:30 PM
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*}
✓ Solution by Maple
Time used: 0.171 (sec). Leaf size: 225
dsolve(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,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}} \\
y \left (x \right ) &= -2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\
y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\
y \left (x \right ) &= -\frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\
y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\
y \left (x \right ) &= \frac {2^{{2}/{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{{1}/{3}}}{2 c_{1}} \\
y \left (x \right ) &= -\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 \left (x \right ) &= \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*}
✓ Solution by Mathematica
Time used: 0.432 (sec). Leaf size: 199
DSolve[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,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*}