Internal problem ID [10056]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1734 (book 6.143).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}+\left (a y+b \right ) y^{2}=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 78
dsolve(2*diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2+(a*y(x)+b)*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ -\sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (-a \,\textit {\_a}^{2}-2 \textit {\_a} b +2 c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {2}\, \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (-a \,\textit {\_a}^{2}-2 \textit {\_a} b +2 c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 6.595 (sec). Leaf size: 1353
DSolve[y[x]^2*(b + a*y[x]) - y'[x]^2 + 2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {i \sqrt {2} \text {$\#$1}^{3/2} \sqrt {2+\frac {4 c_1}{\text {$\#$1} \left (-b+\sqrt {b^2+2 a c_1}\right )}} \sqrt {1-\frac {2 c_1}{\text {$\#$1} \left (b+\sqrt {b^2+2 a c_1}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c_1}{\sqrt {b^2+2 a c_1}-b}}}{\sqrt {\text {$\#$1}}}\right ),\frac {b-\sqrt {b^2+2 a c_1}}{b+\sqrt {b^2+2 a c_1}}\right )}{\sqrt {\frac {c_1}{-b+\sqrt {b^2+2 a c_1}}} \sqrt {-\text {$\#$1} \left (\text {$\#$1}^2 a+2 \text {$\#$1} b-2 c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {i \sqrt {2} \text {$\#$1}^{3/2} \sqrt {2+\frac {4 c_1}{\text {$\#$1} \left (-b+\sqrt {b^2+2 a c_1}\right )}} \sqrt {1-\frac {2 c_1}{\text {$\#$1} \left (b+\sqrt {b^2+2 a c_1}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c_1}{\sqrt {b^2+2 a c_1}-b}}}{\sqrt {\text {$\#$1}}}\right ),\frac {b-\sqrt {b^2+2 a c_1}}{b+\sqrt {b^2+2 a c_1}}\right )}{\sqrt {\frac {c_1}{-b+\sqrt {b^2+2 a c_1}}} \sqrt {-\text {$\#$1} \left (\text {$\#$1}^2 a+2 \text {$\#$1} b-2 c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {i \sqrt {2} \text {$\#$1}^{3/2} \sqrt {2+\frac {4 (-c_1)}{\text {$\#$1} \left (-b+\sqrt {b^2+2 a (-c_1)}\right )}} \sqrt {1-\frac {2 (-c_1)}{\text {$\#$1} \left (b+\sqrt {b^2+2 a (-c_1)}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {-\frac {c_1}{\sqrt {b^2+2 a (-c_1)}-b}}}{\sqrt {\text {$\#$1}}}\right ),\frac {b-\sqrt {b^2+2 a (-c_1)}}{b+\sqrt {b^2+2 a (-c_1)}}\right )}{\sqrt {-\frac {c_1}{-b+\sqrt {b^2+2 a (-c_1)}}} \sqrt {-\text {$\#$1} \left (\text {$\#$1}^2 a+2 \text {$\#$1} b-2 (-1) c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {i \sqrt {2} \text {$\#$1}^{3/2} \sqrt {2+\frac {4 (-c_1)}{\text {$\#$1} \left (-b+\sqrt {b^2+2 a (-c_1)}\right )}} \sqrt {1-\frac {2 (-c_1)}{\text {$\#$1} \left (b+\sqrt {b^2+2 a (-c_1)}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {-\frac {c_1}{\sqrt {b^2+2 a (-c_1)}-b}}}{\sqrt {\text {$\#$1}}}\right ),\frac {b-\sqrt {b^2+2 a (-c_1)}}{b+\sqrt {b^2+2 a (-c_1)}}\right )}{\sqrt {-\frac {c_1}{-b+\sqrt {b^2+2 a (-c_1)}}} \sqrt {-\text {$\#$1} \left (\text {$\#$1}^2 a+2 \text {$\#$1} b-2 (-1) c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {i \sqrt {2} \text {$\#$1}^{3/2} \sqrt {2+\frac {4 c_1}{\text {$\#$1} \left (-b+\sqrt {b^2+2 a c_1}\right )}} \sqrt {1-\frac {2 c_1}{\text {$\#$1} \left (b+\sqrt {b^2+2 a c_1}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c_1}{\sqrt {b^2+2 a c_1}-b}}}{\sqrt {\text {$\#$1}}}\right ),\frac {b-\sqrt {b^2+2 a c_1}}{b+\sqrt {b^2+2 a c_1}}\right )}{\sqrt {\frac {c_1}{-b+\sqrt {b^2+2 a c_1}}} \sqrt {-\text {$\#$1} \left (\text {$\#$1}^2 a+2 \text {$\#$1} b-2 c_1\right )}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {i \sqrt {2} \text {$\#$1}^{3/2} \sqrt {2+\frac {4 c_1}{\text {$\#$1} \left (-b+\sqrt {b^2+2 a c_1}\right )}} \sqrt {1-\frac {2 c_1}{\text {$\#$1} \left (b+\sqrt {b^2+2 a c_1}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c_1}{\sqrt {b^2+2 a c_1}-b}}}{\sqrt {\text {$\#$1}}}\right ),\frac {b-\sqrt {b^2+2 a c_1}}{b+\sqrt {b^2+2 a c_1}}\right )}{\sqrt {\frac {c_1}{-b+\sqrt {b^2+2 a c_1}}} \sqrt {-\text {$\#$1} \left (\text {$\#$1}^2 a+2 \text {$\#$1} b-2 c_1\right )}}\&\right ][x+c_2] \\ \end{align*}