Internal problem ID [12158]
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 61.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
\[ \boxed {\left (-x +y^{2}\right ) y^{\prime }-y=-x^{2}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 318
dsolve((y(x)^2-x)*diff(y(x),x)-y(x)+x^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}+4 x}{2 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (-\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}+4 x \right ) \sqrt {3}-\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}-4 x}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}-4 x \right ) \sqrt {3}-\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}-4 x}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+\left (6 c_{1} -4\right ) x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.856 (sec). Leaf size: 326
DSolve[(y[x]^2-x)*y'[x]-y[x]+x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 x+\sqrt [3]{2} \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x}{4 \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x}{4 \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\ \end{align*}