Internal problem ID [3816]
Internal file name [OUTPUT/3309_Sunday_June_05_2022_09_07_33_AM_6425475/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 20
Problem number: 566.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {\left (1-y x^{2}\right ) y^{\prime }+y^{2} x=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (1-y x^{2}\right ) y^{\prime }+y^{2} x =1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{2} x +1}{1-y x^{2}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 963
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 {4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}}+\left (\left (-c_{1} +80\right ) x^{7}-160 x^{4}+80 x \right ) 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+\left (c_{1}^{2}-80 c_{1} \right ) x^{8}+160 c_{1} x^{5}-80 c_{1} x^{2}}{x^{2} 4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}}+\left (c_{1} x^{4}-4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )} \\ y \left (x \right ) &= \frac {4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )+\left (2 i 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+\left (i-\sqrt {3}\right ) c_{1} x \right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) x}{x^{2} 4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )+\left (2 i 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+\left (i-\sqrt {3}\right ) c_{1} x^{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )} \\ y \left (x \right ) &= \frac {\left (i-\sqrt {3}\right ) 4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}}+\left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) x \left (2 i 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+c_{1} x \left (\sqrt {3}+i\right )\right )}{\left (i-\sqrt {3}\right ) x^{2} 4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}}+\left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) \left (2 i 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+c_{1} x^{4} \left (\sqrt {3}+i\right )\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*}