20.19 problem 566

Internal problem ID [3308]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 20
Problem number: 566.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]

Solve \begin {gather*} \boxed {\left (1-x^{2} y\right ) y^{\prime }-1+x y^{2}=0} \end {gather*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 1583

dsolve((1-x^2*y(x))*diff(y(x),x)-1+x*y(x)^2 = 0,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = -\frac {63 x^{3}-\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}-\frac {63 c_{1} x^{4}}{\left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}}{4 x^{2} \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )}+\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-\frac {63}{4}\right )} \\ y \relax (x ) = -\frac {63 x^{3}+\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{2 \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )}+\frac {63 c_{1} x^{4}}{2 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-2 i \sqrt {3}\, \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{4 x^{2} \left (-\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{8 \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{8 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-\frac {63}{4}+\frac {i \sqrt {3}\, \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ y \relax (x ) = -\frac {63 x^{3}+\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{2 \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )}+\frac {63 c_{1} x^{4}}{2 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}+2 i \sqrt {3}\, \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{4 x^{2} \left (-\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{8 \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{8 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-\frac {63}{4}-\frac {i \sqrt {3}\, \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 24.257 (sec). Leaf size: 476

DSolve[(1-x^2 y[x])y'[x]-1+x y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}{-1+6 c_1}-\frac {x^2}{\sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}+x \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}{-2+12 c_1}+\frac {\left (1+i \sqrt {3}\right ) x^2}{2 \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}+x \\ y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}{-2+12 c_1}+\frac {\left (1-i \sqrt {3}\right ) x^2}{2 \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}+x \\ y(x)\to x \\ \end{align*}