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: 0.25 (sec). Leaf size: 231
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 ) &= 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}} \\ y \left (x \right ) &= -2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4 c_{1}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{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*}