Internal problem ID [11194]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter 2, differential equations of the first order and the first degree. Article 19.
Summary. Page 29
Problem number: Ex 25.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {2 y^{2} x^{3}-y+\left (2 y^{3} x^{2}-x \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 361
dsolve((2*x^3*y(x)^2-y(x))+(2*x^2*y(x)^3-x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\left (-{\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {2}{3}}+x^{2} 12^{\frac {1}{3}} \left (x^{2}-c_{1} \right )\right ) 12^{\frac {1}{3}}}{6 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} x^{2} \left (x^{2}-c_{1} \right )\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} x^{2} \left (x^{2}-c_{1} \right ) \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right )\right )}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 46.278 (sec). Leaf size: 358
DSolve[(2*x^3*y[x]^2-y[x])+(2*x^2*y[x]^3-x)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (-x^3+c_1 x\right )}{\sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}+\frac {\sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}{3 \sqrt [3]{2} x} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-c_1 x\right )}{2^{2/3} \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}{6 \sqrt [3]{2} x} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-c_1 x\right )}{2^{2/3} \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}{6 \sqrt [3]{2} x} \\ \end{align*}