Internal problem ID [9392]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1814 (book 6.223).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {\left (b +a \left (\sin ^{2}\relax (y)\right )\right ) y^{\prime \prime }+a \left (y^{\prime }\right )^{2} \cos \relax (y) \sin \relax (y)+A y \left (c +a \left (\sin ^{2}\relax (y)\right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 139
dsolve((b+a*sin(y(x))^2)*diff(diff(y(x),x),x)+a*diff(y(x),x)^2*cos(y(x))*sin(y(x))+A*y(x)*(c+a*sin(y(x))^2)=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}-\frac {2 \left (b +a \left (\sin ^{2}\left (\textit {\_a} \right )\right )\right )}{\sqrt {-2 \left (b +a \left (\sin ^{2}\left (\textit {\_a} \right )\right )\right ) \left (-2 A a \textit {\_a} \sin \left (\textit {\_a} \right ) \cos \left (\textit {\_a} \right )+A a \left (\sin ^{2}\left (\textit {\_a} \right )\right )+A a \,\textit {\_a}^{2}+2 A c \,\textit {\_a}^{2}-2 c_{1}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {2 b +2 a \left (\sin ^{2}\left (\textit {\_a} \right )\right )}{\sqrt {-2 \left (b +a \left (\sin ^{2}\left (\textit {\_a} \right )\right )\right ) \left (-2 A a \textit {\_a} \sin \left (\textit {\_a} \right ) \cos \left (\textit {\_a} \right )+A a \left (\sin ^{2}\left (\textit {\_a} \right )\right )+A a \,\textit {\_a}^{2}+2 A c \,\textit {\_a}^{2}-2 c_{1}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 16.71 (sec). Leaf size: 176
DSolve[A*(c + a*Sin[y[x]]^2)*y[x] + a*Cos[y[x]]*Sin[y[x]]*y'[x]^2 + (b + a*Sin[y[x]]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2} \sqrt {\cos (2 K[1]) a-a-2 b}}{\sqrt {2 a A K[1]^2+4 A c K[1]^2-2 a A \sin (2 K[1]) K[1]+2 c_1-a A \cos (2 K[1])}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2} \sqrt {\cos (2 K[2]) a-a-2 b}}{\sqrt {2 a A K[2]^2+4 A c K[2]^2-2 a A \sin (2 K[2]) K[2]+2 c_1-a A \cos (2 K[2])}}dK[2]\&\right ][x+c_2] \\ \end{align*}