Internal problem ID [12151]
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]
\[ \boxed {-y+\left (x^{2} y^{2}+x \right ) y^{\prime }=-x^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 346
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 {2^{\frac {1}{3}} \left (-\frac {2^{\frac {1}{3}} {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}}{2}+x \right )}{{\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+2 i \sqrt {3}\, x -2 x \right ) 2^{\frac {1}{3}}}{4 {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) 2^{\frac {2}{3}} {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+2 \left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} x}{4 {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 56.22 (sec). Leaf size: 400
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+\left (-6 x^4+6 c_1 x^3+2 \sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}\right ){}^{2/3}}{2 x \sqrt [3]{-3 x^4+3 c_1 x^3+\sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \left (-6 x^4+6 c_1 x^3+2 \sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x}{4 x \sqrt [3]{-3 x^4+3 c_1 x^3+\sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}}} \\ y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \left (-6 x^4+6 c_1 x^3+2 \sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}\right ){}^{2/3}+\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x}{4 x \sqrt [3]{-3 x^4+3 c_1 x^3+\sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}}} \\ \end{align*}