Internal problem ID [9324]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1746 (book 6.155).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {2 \left (y-a \right ) y^{\prime \prime }+\left (y^{\prime }\right )^{2}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 293
dsolve(2*(y(x)-a)*diff(diff(y(x),x),x)+diff(y(x),x)^2+1=0,y(x), singsol=all)
\begin{align*} -\sqrt {-y \relax (x )^{2}+\left (2 a +c_{1}\right ) y \relax (x )-a \left (c_{1}+a \right )}+\arctan \left (\frac {y \relax (x )-a -\frac {c_{1}}{2}}{\sqrt {-y \relax (x )^{2}+\left (2 a +c_{1}\right ) y \relax (x )-a \left (c_{1}+a \right )}}\right ) a +\frac {\arctan \left (\frac {y \relax (x )-a -\frac {c_{1}}{2}}{\sqrt {-y \relax (x )^{2}+\left (2 a +c_{1}\right ) y \relax (x )-a \left (c_{1}+a \right )}}\right ) c_{1}}{2}-a \arctan \left (\frac {y \relax (x )-a -\frac {c_{1}}{2}}{\sqrt {-y \relax (x )^{2}+\left (2 a +c_{1}\right ) y \relax (x )+a \left (-c_{1}-a \right )}}\right )-x -c_{2} = 0 \\ \sqrt {-y \relax (x )^{2}+\left (2 a +c_{1}\right ) y \relax (x )-a \left (c_{1}+a \right )}-\arctan \left (\frac {y \relax (x )-a -\frac {c_{1}}{2}}{\sqrt {-y \relax (x )^{2}+\left (2 a +c_{1}\right ) y \relax (x )-a \left (c_{1}+a \right )}}\right ) a -\frac {\arctan \left (\frac {y \relax (x )-a -\frac {c_{1}}{2}}{\sqrt {-y \relax (x )^{2}+\left (2 a +c_{1}\right ) y \relax (x )-a \left (c_{1}+a \right )}}\right ) c_{1}}{2}+a \arctan \left (\frac {y \relax (x )-a -\frac {c_{1}}{2}}{\sqrt {-y \relax (x )^{2}+\left (2 a +c_{1}\right ) y \relax (x )+a \left (-c_{1}-a \right )}}\right )-x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.539 (sec). Leaf size: 203
DSolve[1 + y'[x]^2 + 2*(-a + y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-\text {$\#$1}} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}+i \sqrt {2} e^{2 c_1} \log \left (\sqrt {2 a-2 \text {$\#$1}}-i \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}\right )}{2 \sqrt {2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {a-\text {$\#$1}} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}+i \sqrt {2} e^{2 c_1} \log \left (\sqrt {2 a-2 \text {$\#$1}}-i \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}\right )}{2 \sqrt {2}}\&\right ][x+c_2] \\ \end{align*}