29.34.6 problem 1001
Internal
problem
ID
[5577]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1001
Date
solved
:
Monday, January 27, 2025 at 12:10:53 PM
CAS
classification
:
[_rational]
\begin{align*} x y^{2} {y^{\prime }}^{2}+\left (a -x^{3}-y^{3}\right ) y^{\prime }+x^{2} y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.255 (sec). Leaf size: 247
dsolve(x*y(x)^2*diff(y(x),x)^2+(a-x^3-y(x)^3)*diff(y(x),x)+x^2*y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \left (x^{3}+a -2 x \sqrt {a x}\right )^{{1}/{3}} \\
y \left (x \right ) &= \left (x^{3}+a +2 x \sqrt {a x}\right )^{{1}/{3}} \\
y \left (x \right ) &= -\frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\
y \left (x \right ) &= \frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\
y \left (x \right ) &= -\frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\
y \left (x \right ) &= \frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\
y \left (x \right ) &= 0 \\
\int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} +\frac {\ln \left (x \right )}{2}-c_{1} &= 0 \\
\int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} -\frac {\ln \left (x \right )}{2}-c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.684 (sec). Leaf size: 194
DSolve[x y[x]^2 (D[y[x],x])^2 +(a-x^3-y[x]^3) D[y[x],x]+x^2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{a+(-1+c_1) x^3}}{\sqrt [3]{1-\frac {1}{c_1}}} \\
y(x)\to 0 \\
y(x)\to \sqrt [3]{\left (\sqrt {a}-x^{3/2}\right )^2} \\
y(x)\to -\sqrt [3]{-1} \sqrt [3]{\left (\sqrt {a}-x^{3/2}\right )^2} \\
y(x)\to (-1)^{2/3} \sqrt [3]{\left (\sqrt {a}-x^{3/2}\right )^2} \\
y(x)\to \sqrt [3]{\left (\sqrt {a}+x^{3/2}\right )^2} \\
y(x)\to -\sqrt [3]{-1} \sqrt [3]{\left (\sqrt {a}+x^{3/2}\right )^2} \\
y(x)\to (-1)^{2/3} \sqrt [3]{\left (\sqrt {a}+x^{3/2}\right )^2} \\
\end{align*}