Internal problem ID [9421]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1842.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime \prime }+x \left (y-1\right ) y^{\prime \prime }+x \left (y^{\prime }\right )^{2}+\left (1-y\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 190
dsolve(x^2*diff(diff(diff(y(x),x),x),x)+x*(-1+y(x))*diff(diff(y(x),x),x)+x*diff(y(x),x)^2+(1-y(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ \ln \relax (x )+2 \left (\int _{}^{y \relax (x )}\frac {1}{2 \RootOf \left (-2 \BesselY \left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {4+c_{1}}\, c_{2}+2 \BesselY \left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_{2} \textit {\_h} -4 \BesselY \left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_{2}+2 \sqrt {2}\, \BesselY \left (\frac {\sqrt {4+c_{1}}}{2}+1, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_{2} \textit {\_Z} +2 \sqrt {2}\, \BesselJ \left (\frac {\sqrt {4+c_{1}}}{2}+1, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \textit {\_Z} -2 \BesselJ \left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {4+c_{1}}+2 \BesselJ \left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \textit {\_h} -4 \BesselJ \left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right )\right )^{2}+\textit {\_h}^{2}-c_{1}-4 \textit {\_h}}d \textit {\_h} \right )-c_{3} = 0 \]
✓ Solution by Mathematica
Time used: 0.123 (sec). Leaf size: 245
DSolve[(1 - y[x])*y'[x] + x*y'[x]^2 + x*(-1 + y[x])*y''[x] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_3 \left (i \sqrt {c_1} x J_{\frac {\sqrt {c_2+2}}{\sqrt {2}}+1}\left (-\frac {1}{2} i x \sqrt {c_1}\right )+\left (2+\sqrt {2} \sqrt {2+c_2}\right ) J_{\frac {\sqrt {c_2+2}}{\sqrt {2}}}\left (-\frac {1}{2} i x \sqrt {c_1}\right )\right )+i \sqrt {c_1} x Y_{\frac {\sqrt {c_2+2}}{\sqrt {2}}+1}\left (-\frac {1}{2} i x \sqrt {c_1}\right )+\left (2+\sqrt {2} \sqrt {2+c_2}\right ) Y_{\frac {\sqrt {c_2+2}}{\sqrt {2}}}\left (-\frac {1}{2} i x \sqrt {c_1}\right )}{c_3 J_{\frac {\sqrt {c_2+2}}{\sqrt {2}}}\left (-\frac {1}{2} i x \sqrt {c_1}\right )+Y_{\frac {\sqrt {c_2+2}}{\sqrt {2}}}\left (-\frac {1}{2} i x \sqrt {c_1}\right )} \\ \end{align*}