60.1.506 problem 509
Internal
problem
ID
[10520]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
509
Date
solved
:
Monday, January 27, 2025 at 08:44:45 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} 9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }-4 x^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 225
dsolve(9*y(x)^4*(x^2-1)*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 &= 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*}
✓ Solution by Mathematica
Time used: 0.433 (sec). Leaf size: 199
DSolve[-4*x^2 - 6*x*y[x]^5*D[y[x],x] + 9*(-1 + x^2)*y[x]^4*D[y[x],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*}