Internal problem ID [10136]
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]]
\[ \boxed {\left (b +a \sin \left (y\right )^{2}\right ) y^{\prime \prime }+a {y^{\prime }}^{2} \cos \left (y\right ) \sin \left (y\right )+A y \left (c +a \sin \left (y\right )^{2}\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 140
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*} \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {b +a \sin \left (\textit {\_a} \right )^{2}}{\sqrt {-\left (A a \sin \left (\textit {\_a} \right )^{2}-2 A a \textit {\_a} \cos \left (\textit {\_a} \right ) \sin \left (\textit {\_a} \right )+\textit {\_a}^{2} \left (a +2 c \right ) A -2 c_{1} \right ) \left (b +a \sin \left (\textit {\_a} \right )^{2}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {b +a \sin \left (\textit {\_a} \right )^{2}}{\sqrt {-\left (A a \sin \left (\textit {\_a} \right )^{2}-2 A a \textit {\_a} \cos \left (\textit {\_a} \right ) \sin \left (\textit {\_a} \right )+\textit {\_a}^{2} \left (a +2 c \right ) A -2 c_{1} \right ) \left (b +a \sin \left (\textit {\_a} \right )^{2}\right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 56.768 (sec). Leaf size: 530
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 {$\#$1}}-\frac {\sqrt {2} \sqrt {\cos (2 K[1]) a-a-2 b}}{\sqrt {2 a A K[1]^2+4 A c K[1]^2-2 a A \sin (2 K[1]) K[1]+2 c_1-a A \cos (2 K[1])}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2} \sqrt {\cos (2 K[2]) a-a-2 b}}{\sqrt {2 a A K[2]^2+4 A c K[2]^2-2 a A \sin (2 K[2]) K[2]+2 c_1-a A \cos (2 K[2])}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2} \sqrt {\cos (2 K[1]) a-a-2 b}}{\sqrt {2 a A K[1]^2+4 A c K[1]^2-2 a A \sin (2 K[1]) K[1]+2 (-1) c_1-a A \cos (2 K[1])}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2} \sqrt {\cos (2 K[1]) a-a-2 b}}{\sqrt {2 a A K[1]^2+4 A c K[1]^2-2 a A \sin (2 K[1]) K[1]+2 c_1-a A \cos (2 K[1])}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2} \sqrt {\cos (2 K[2]) a-a-2 b}}{\sqrt {2 a A K[2]^2+4 A c K[2]^2-2 a A \sin (2 K[2]) K[2]+2 (-1) c_1-a A \cos (2 K[2])}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$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*}