Internal problem ID [10803]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 1, First-Order Differential Equations. Problems page 88
Problem number: Problem 54.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {x^{2}-y+\left (y^{2} x^{2}+x \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 508
dsolve((x^2-y(x))+(x^2*y(x)^2+x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{2 x}-\frac {2}{{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}} \\ y \left (x \right ) = -\frac {{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{4 x}+\frac {1}{{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{2 x}+\frac {2}{{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2} \\ y \left (x \right ) = -\frac {{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{4 x}+\frac {1}{{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}{2 x}+\frac {2}{{\left (\left (-12 c_{1} x -12 x^{2}+4 \sqrt {\frac {9 c_{1}^{2} x^{3}+18 c_{1} x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 48.634 (sec). Leaf size: 319
DSolve[(x^2-y[x])+(x^2*y[x]^2+x)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 \sqrt [3]{2} x+2^{2/3} \left (3 x^3 (-x+c_1)+\sqrt {x^3 \left (4+9 x^3 (x-c_1){}^2\right )}\right ){}^{2/3}}{2 x \sqrt [3]{3 x^3 (-x+c_1)+\sqrt {x^3 \left (4+9 x^3 (x-c_1){}^2\right )}}} \\ y(x)\to \frac {2 \sqrt [3]{-2} x+(-2)^{2/3} \left (3 x^3 (-x+c_1)+\sqrt {x^3 \left (4+9 x^3 (x-c_1){}^2\right )}\right ){}^{2/3}}{2 x \sqrt [3]{3 x^3 (-x+c_1)+\sqrt {x^3 \left (4+9 x^3 (x-c_1){}^2\right )}}} \\ y(x)\to \frac {-\sqrt [3]{-2} \left (3 x^3 (-x+c_1)+\sqrt {x^3 \left (4+9 x^3 (x-c_1){}^2\right )}\right ){}^{2/3}-i \sqrt {3} x+x}{2^{2/3} x \sqrt [3]{3 x^3 (-x+c_1)+\sqrt {x^3 \left (4+9 x^3 (x-c_1){}^2\right )}}} \\ \end{align*}