Internal problem ID [7444]
Internal file name [OUTPUT/6411_Sunday_June_05_2022_04_47_20_PM_43215893/index.tex]
Book: Second order enumerated odes
Section: section 2
Problem number: 3.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y^{2} {y^{\prime }}^{2}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville <- 2nd_order Liouville successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 62
dsolve(diff(y(x),x$2)+(1-x)*diff(y(x),x)+y(x)^2*diff(y(x),x)^2=0,y(x), singsol=all)
\[ c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -1\right )}{2}\right )-c_{2} +\frac {2 \,3^{\frac {5}{6}} y \left (x \right ) \pi }{9 \Gamma \left (\frac {2}{3}\right ) \left (-y \left (x \right )^{3}\right )^{\frac {1}{3}}}-\frac {y \left (x \right ) \Gamma \left (\frac {1}{3}, -\frac {y \left (x \right )^{3}}{3}\right ) 3^{\frac {1}{3}}}{3 \left (-y \left (x \right )^{3}\right )^{\frac {1}{3}}} = 0 \]
✓ Solution by Mathematica
Time used: 0.374 (sec). Leaf size: 67
DSolve[y''[x]+(1-x)*y'[x]+y[x]^2*(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \Gamma \left (\frac {1}{3},-\frac {\text {$\#$1}^3}{3}\right )}{3^{2/3} \sqrt [3]{-\text {$\#$1}^3}}\&\right ]\left [c_2-\sqrt {\frac {\pi }{2 e}} c_1 \text {erfi}\left (\frac {x-1}{\sqrt {2}}\right )\right ] \]