Internal problem ID [6688]
Book: Second order enumerated odes
Section: section 2
Problem number: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y^{2} \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 64
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)
\[ c_{1} \erf \left (\frac {i \sqrt {2}\, x}{2}-\frac {i \sqrt {2}}{2}\right )-c_{2}+\frac {2 \,3^{\frac {5}{6}} y \relax (x ) \pi }{9 \Gamma \left (\frac {2}{3}\right ) \left (-y \relax (x )^{3}\right )^{\frac {1}{3}}}-\frac {y \relax (x ) \Gamma \left (\frac {1}{3}, -\frac {y \relax (x )^{3}}{3}\right )}{\left (-9 y \relax (x )^{3}\right )^{\frac {1}{3}}} = 0 \]
✓ Solution by Mathematica
Time used: 0.3 (sec). Leaf size: 67
DSolve[y''[x]+(1-x)*y'[x]+y[x]^2*(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \text {Gamma}\left (\frac {1}{3},-\frac {\text {$\#$1}^3}{3}\right )}{3^{2/3} \sqrt [3]{-\text {$\#$1}^3}}\&\right ]\left [c_2-\sqrt {\frac {\pi }{2 e}} c_1 \text {Erfi}\left (\frac {x-1}{\sqrt {2}}\right )\right ] \\ \end{align*}