Internal
problem
ID
[8092]
Book
:
Second
order
enumerated
odes
Section
:
section
2
Problem
number
:
3
Date
solved
:
Tuesday, October 22, 2024 at 03:00:18 PM
CAS
classification
:
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y^{2} {y^{\prime }}^{2}&=0 \end{align*}
2.3.3 Maple dsolve solution
Solving time : 0.012
(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)
\[
c_1 \,\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -1\right )}{2}\right )-c_2 +\frac {2 \,3^{{5}/{6}} y \pi }{9 \Gamma \left (\frac {2}{3}\right ) \left (-y^{3}\right )^{{1}/{3}}}-\frac {y \Gamma \left (\frac {1}{3}, -\frac {y^{3}}{3}\right ) 3^{{1}/{3}}}{3 \left (-y^{3}\right )^{{1}/{3}}} = 0
\]
2.3.4 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]
\[
y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \Gamma \left (\frac {1}{3},-\frac {\text {$\#$1}^3}{3}\right )}{3^{2/3} \sqrt [3]{-\text {$\#$1}^3}}\&\right ]\left [c_2-\sqrt {\frac {\pi }{2 e}} c_1 \text {erfi}\left (\frac {x-1}{\sqrt {2}}\right )\right ]
\]