Internal problem ID [7850]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 270.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {\left (y^{2}-x \right ) y^{\prime }-y+x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 402
dsolve((y(x)^2-x)*diff(y(x),x)-y(x)+x^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 x}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 x}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 c_{1} x^{3}-4 x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.014 (sec). Leaf size: 304
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 {\left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3} \text {Root}\left [\text {$\#$1}^3+16\&,2\right ]+\left (2+2 i \sqrt {3}\right ) x}{2\ 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 {\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}-i \sqrt {3} x+x}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \\ \end{align*}