20.19 problem 566

Internal problem ID [3816]

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`]]

\[ \boxed {\left (1-x^{2} y\right ) y^{\prime }+y^{2} x=1} \]

✓ Solution by Maple

Time used: 0.0 (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 \left (x \right ) = -\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 \left (x \right ) = -\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 \left (x \right ) = -\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: 36.012 (sec). Leaf size: 506

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 (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 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 (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 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 (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 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 (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 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 (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 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 (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}+x y(x)\to x \end{align*}