Internal problem ID [10054]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1723 (book 6.132).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}-2 a y \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=1} \]
✓ Solution by Maple
Time used: 0.782 (sec). Leaf size: 116
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-1-2*a*y(x)*(diff(y(x),x)^2+1)^(3/2)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -i x +c_{1} y \left (x \right ) = i x +c_{1} \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2} a +c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-2 c_{1} \textit {\_a}^{2} a -c_{1}^{2}+\textit {\_a}^{2}}}d \textit {\_a} -x -c_{2} = 0 \int _{}^{y \left (x \right )}-\frac {\textit {\_a}^{2} a +c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-2 c_{1} \textit {\_a}^{2} a -c_{1}^{2}+\textit {\_a}^{2}}}d \textit {\_a} -x -c_{2} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 5.652 (sec). Leaf size: 2181
DSolve[-1 - y'[x]^2 - 2*a*y[x]*(1 + y'[x]^2)^(3/2) + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\frac {2 \text {$\#$1}^2 a^2}{-2 a c_1+\sqrt {1-4 a c_1}+1}} \sqrt {1+\frac {2 \text {$\#$1}^2 a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \left (\left (-2 a c_1+\sqrt {1-4 a c_1}+1\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right )|-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )-\left (1+\sqrt {1-4 a c_1}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right ),-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )\right )}{2 \sqrt {2} a \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \sqrt {\text {$\#$1}^4 a^2+\text {$\#$1}^2 (-1+2 a c_1)+c_1{}^2}}\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\frac {2 \text {$\#$1}^2 a^2}{-2 a c_1+\sqrt {1-4 a c_1}+1}} \sqrt {1+\frac {2 \text {$\#$1}^2 a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \left (\left (-2 a c_1+\sqrt {1-4 a c_1}+1\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right )|-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )-\left (1+\sqrt {1-4 a c_1}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right ),-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )\right )}{2 \sqrt {2} a \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \sqrt {\text {$\#$1}^4 a^2+\text {$\#$1}^2 (-1+2 a c_1)+c_1{}^2}}\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\frac {2 \text {$\#$1}^2 a^2}{-2 a (-c_1)+\sqrt {1-4 a (-c_1)}+1}} \sqrt {1+\frac {2 \text {$\#$1}^2 a^2}{2 a (-c_1)+\sqrt {1-4 a (-c_1)}-1}} \left (\left (-2 a (-c_1)+\sqrt {1-4 a (-c_1)}+1\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 (-c_1) a+\sqrt {1-4 a (-c_1)}-1}} \text {$\#$1}\right )|-\frac {2 (-c_1) a+\sqrt {1-4 a (-c_1)}-1}{-2 (-c_1) a+\sqrt {1-4 a (-c_1)}+1}\right )-\left (1+\sqrt {1-4 a (-c_1)}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 (-c_1) a+\sqrt {1-4 a (-c_1)}-1}} \text {$\#$1}\right ),-\frac {2 (-c_1) a+\sqrt {1-4 a (-c_1)}-1}{-2 (-c_1) a+\sqrt {1-4 a (-c_1)}+1}\right )\right )}{2 \sqrt {2} a \sqrt {\frac {a^2}{2 a (-c_1)+\sqrt {1-4 a (-c_1)}-1}} \sqrt {\text {$\#$1}^4 a^2+\text {$\#$1}^2 (-1+2 a (-c_1))+(-c_1){}^2}}\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\frac {2 \text {$\#$1}^2 a^2}{-2 a (-c_1)+\sqrt {1-4 a (-c_1)}+1}} \sqrt {1+\frac {2 \text {$\#$1}^2 a^2}{2 a (-c_1)+\sqrt {1-4 a (-c_1)}-1}} \left (\left (-2 a (-c_1)+\sqrt {1-4 a (-c_1)}+1\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 (-c_1) a+\sqrt {1-4 a (-c_1)}-1}} \text {$\#$1}\right )|-\frac {2 (-c_1) a+\sqrt {1-4 a (-c_1)}-1}{-2 (-c_1) a+\sqrt {1-4 a (-c_1)}+1}\right )-\left (1+\sqrt {1-4 a (-c_1)}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 (-c_1) a+\sqrt {1-4 a (-c_1)}-1}} \text {$\#$1}\right ),-\frac {2 (-c_1) a+\sqrt {1-4 a (-c_1)}-1}{-2 (-c_1) a+\sqrt {1-4 a (-c_1)}+1}\right )\right )}{2 \sqrt {2} a \sqrt {\frac {a^2}{2 a (-c_1)+\sqrt {1-4 a (-c_1)}-1}} \sqrt {\text {$\#$1}^4 a^2+\text {$\#$1}^2 (-1+2 a (-c_1))+(-c_1){}^2}}\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\frac {2 \text {$\#$1}^2 a^2}{-2 a c_1+\sqrt {1-4 a c_1}+1}} \sqrt {1+\frac {2 \text {$\#$1}^2 a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \left (\left (-2 a c_1+\sqrt {1-4 a c_1}+1\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right )|-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )-\left (1+\sqrt {1-4 a c_1}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right ),-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )\right )}{2 \sqrt {2} a \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \sqrt {\text {$\#$1}^4 a^2+\text {$\#$1}^2 (-1+2 a c_1)+c_1{}^2}}\&\right ][x+c_2] y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\frac {2 \text {$\#$1}^2 a^2}{-2 a c_1+\sqrt {1-4 a c_1}+1}} \sqrt {1+\frac {2 \text {$\#$1}^2 a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \left (\left (-2 a c_1+\sqrt {1-4 a c_1}+1\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right )|-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )-\left (1+\sqrt {1-4 a c_1}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \text {$\#$1}\right ),-\frac {2 a c_1+\sqrt {1-4 a c_1}-1}{-2 a c_1+\sqrt {1-4 a c_1}+1}\right )\right )}{2 \sqrt {2} a \sqrt {\frac {a^2}{2 a c_1+\sqrt {1-4 a c_1}-1}} \sqrt {\text {$\#$1}^4 a^2+\text {$\#$1}^2 (-1+2 a c_1)+c_1{}^2}}\&\right ][x+c_2] \end{align*}