60.1.501 problem 504
Internal
problem
ID
[10515]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
504
Date
solved
:
Monday, January 27, 2025 at 08:41:57 PM
CAS
classification
:
[_rational]
\begin{align*} x y^{2} {y^{\prime }}^{2}-\left (y^{3}+x^{3}-a \right ) y^{\prime }+x^{2} y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.360 (sec). Leaf size: 247
dsolve(x*y(x)^2*diff(y(x),x)^2-(y(x)^3+x^3-a)*diff(y(x),x)+x^2*y(x)=0,y(x), singsol=all)
\begin{align*}
y &= \left (x^{3}+a -2 x \sqrt {a x}\right )^{{1}/{3}} \\
y &= \left (x^{3}+a +2 x \sqrt {a x}\right )^{{1}/{3}} \\
y &= -\frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\
y &= \frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\
y &= -\frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\
y &= \frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\
y &= 0 \\
\int _{\textit {\_b}}^{y}\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}\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: 0.749 (sec). Leaf size: 194
DSolve[x^2*y[x] - (-a + x^3 + y[x]^3)*D[y[x],x] + x*y[x]^2*D[y[x],x]^2==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*}