Internal problem ID [9961]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1638 (6.48).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime }+a {y^{\prime }}^{2}+b \sin \left (y\right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 211
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)^2+b*sin(y(x))=0,y(x), singsol=all)
\begin{align*} 4 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_{1} +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} \right ) a^{2}+\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_{1} +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} -c_{2} -x &= 0 \\ -4 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_{1} +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} \right ) a^{2}-\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_{1} +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} \right )-c_{2} -x &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 10.587 (sec). Leaf size: 444
DSolve[b*Sin[y[x]] + a*y'[x]^2 + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[1]} c_1 a^2-4 b \sin (K[1]) a+e^{-2 a K[1]} c_1+2 b \cos (K[1])}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[2]} c_1 a^2-4 b \sin (K[2]) a+e^{-2 a K[2]} c_1+2 b \cos (K[2])}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[1]} (-c_1) a^2-4 b \sin (K[1]) a+e^{-2 a K[1]} (-c_1)+2 b \cos (K[1])}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[1]} c_1 a^2-4 b \sin (K[1]) a+e^{-2 a K[1]} c_1+2 b \cos (K[1])}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[2]} (-c_1) a^2-4 b \sin (K[2]) a+e^{-2 a K[2]} (-c_1)+2 b \cos (K[2])}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {4 a^2+1}}{\sqrt {4 e^{-2 a K[2]} c_1 a^2-4 b \sin (K[2]) a+e^{-2 a K[2]} c_1+2 b \cos (K[2])}}dK[2]\&\right ][x+c_2] \\ \end{align*}