Internal problem ID [9288]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1710 (book 6.119).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y-\left (y^{\prime }\right )^{2}-\left (a y-1\right ) y^{\prime }+2 a^{2} y^{2}-2 b^{2} y^{3}+a y=0} \end {gather*}
✗ Solution by Maple
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-(a*y(x)-1)*diff(y(x),x)+2*a^2*y(x)^2-2*b^2*y(x)^3+a*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 48.767 (sec). Leaf size: 540
DSolve[a*y[x] + 2*a^2*y[x]^2 - 2*b^2*y[x]^3 - (-1 + a*y[x])*y'[x] - y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2 a}+e^{2 a x} \left (\frac {e^{-2 a x} \left (c_1 \left (a^{3/2}-\sqrt {a^3+2 b^2}\right ) \Gamma \left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) J_{-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}}\left (\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )-2 c_1 \Gamma \left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \sqrt {a b^2 c_2 e^{2 a x}} J_{1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}}\left (\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )+\Gamma \left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1\right ) \left (\left (a^{3/2}+\sqrt {a^3+2 b^2}\right ) J_{\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}}\left (\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )-2 \sqrt {a b^2 c_2 e^{2 a x}} J_{\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1}\left (\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )\right )\right ){}^2}{4 a b^2 \left (c_1 \Gamma \left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) J_{-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}}\left (\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )+\Gamma \left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1\right ) J_{\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}}\left (\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )\right ){}^2}+c_2\right ) \\ \end{align*}