2.2.3 problem 3

Internal problem ID [8538]
Book : Second order enumerated odes
Section : section 2
Problem number : 3
Date solved : Sunday, November 10, 2024 at 09:12:29 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Solve

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

Maple step by step solution

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
<- 2nd_order Liouville successful`
 

Maple dsolve solution

Solving time : 0.015 (sec)
Leaf size : 61

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

\[ c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -1\right )}{2}\right )-c_{2} +\frac {2 \,3^{{5}/{6}} y \pi }{9 \Gamma \left (\frac {2}{3}\right ) \left (-y^{3}\right )^{{1}/{3}}}-\frac {y \Gamma \left (\frac {1}{3}, -\frac {y^{3}}{3}\right ) 3^{{1}/{3}}}{3 \left (-y^{3}\right )^{{1}/{3}}} = 0 \]

Mathematica DSolve solution

Solving time : 0.512 (sec)
Leaf size : 67

DSolve[{D[y[x],{x,2}]+(1-x)*D[y[x],x]+y[x]^2*(D[y[x],x])^2==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \Gamma \left (\frac {1}{3},-\frac {\text {$\#$1}^3}{3}\right )}{3^{2/3} \sqrt [3]{-\text {$\#$1}^3}}\&\right ]\left [c_2-\sqrt {\frac {\pi }{2 e}} c_1 \text {erfi}\left (\frac {x-1}{\sqrt {2}}\right )\right ] \]