Internal problem ID [9297]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1718.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y+a \left (y^{\prime }\right )^{2}+b y y^{\prime }+c y^{2}+d y^{-a +1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.072 (sec). Leaf size: 145
dsolve(diff(diff(y(x),x),x)*y(x)+a*diff(y(x),x)^2+b*y(x)*diff(y(x),x)+c*y(x)^2+d*y(x)^(1-a)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-\frac {x \sqrt {-4 c a +b^{2}-4 c}}{2 \left (a +1\right )}} {\mathrm e}^{-\frac {b x}{2 \left (a +1\right )}} \left (\frac {c^{2} \left (-4 c a +b^{2}-4 c \right )}{\left ({\mathrm e}^{x \sqrt {-4 c a +b^{2}-4 c}} c_{1} a c +{\mathrm e}^{x \sqrt {-4 c a +b^{2}-4 c}} c_{1} c +d \,{\mathrm e}^{\frac {\left (b +\sqrt {-4 c a +b^{2}-4 c}\right ) x}{2}} \sqrt {-4 c a +b^{2}-4 c}-c_{2} c a -c_{2} c \right )^{2}}\right )^{-\frac {1}{2 \left (a +1\right )}} \]
✓ Solution by Mathematica
Time used: 0.96 (sec). Leaf size: 161
DSolve[c*y[x]^2 + d*y[x]^(1 - a) + b*y[x]*y'[x] + a*y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (\frac {e^{-\frac {1}{2} x \left (\sqrt {b^2-4 (a+1) c}+b\right )} \left (d \left (b^2-4 (a+1) c\right ) e^{\frac {1}{2} x \left (\sqrt {b^2-4 (a+1) c}+b\right )}+c c_2 \left (4 (a+1) c-b^2\right ) e^{x \sqrt {b^2-4 (a+1) c}}-a c c_1 \sqrt {b^2-4 (a+1) c}-c c_1 \sqrt {b^2-4 (a+1) c}\right )}{c \left (4 (a+1) c-b^2\right )}\right ){}^{\frac {1}{a+1}} \\ \end{align*}