Internal problem ID [8844]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 509.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }=4 x^{2}} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 245
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 \left (x \right ) = \left (-4 x^{2}+4\right )^{\frac {1}{6}} y \left (x \right ) = -\left (-4 x^{2}+4\right )^{\frac {1}{6}} y \left (x \right ) = \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-4 x^{2}+4\right )^{\frac {1}{6}} y \left (x \right ) = \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-4 x^{2}+4\right )^{\frac {1}{6}} y \left (x \right ) = \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-4 x^{2}+4\right )^{\frac {1}{6}} y \left (x \right ) = \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-4 x^{2}+4\right )^{\frac {1}{6}} y \left (x \right ) = \frac {{\left (\left (-16 c_{1}^{2}+4 x^{2}-4\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{2 c_{1}} y \left (x \right ) = -\frac {{\left (\left (-16 c_{1}^{2}+4 x^{2}-4\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{4 c_{1}}-\frac {i \sqrt {3}\, {\left (\left (-16 c_{1}^{2}+4 x^{2}-4\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{4 c_{1}} y \left (x \right ) = -\frac {{\left (\left (-16 c_{1}^{2}+4 x^{2}-4\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{4 c_{1}}+\frac {i \sqrt {3}\, {\left (\left (-16 c_{1}^{2}+4 x^{2}-4\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{4 c_{1}} \end{align*}
✓ Solution by Mathematica
Time used: 0.429 (sec). Leaf size: 199
DSolve[-4*x^2 - 6*x*y[x]^5*y'[x] + 9*(-1 + x^2)*y[x]^4*y'[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*}