Internal problem ID [10180]
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]
Solve \begin {gather*} \boxed {2 x^{3} y^{2}-y+\left (2 x^{2} y^{3}-x \right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 522
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 \relax (x ) = \frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {6 \left (\frac {x^{2}}{3}-\frac {c_{1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}+\frac {3 \left (\frac {x^{2}}{3}-\frac {c_{1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {6 \left (\frac {x^{2}}{3}-\frac {c_{1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}+\frac {3 \left (\frac {x^{2}}{3}-\frac {c_{1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {6 \left (\frac {x^{2}}{3}-\frac {c_{1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 x^{6} c_{1}+36 x^{4} c_{1}^{2}-12 x^{2} c_{1}^{3}+81}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 28.406 (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*}