Internal problem ID [10068]
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]]
\[ \boxed {2 \left (y-a \right ) y^{\prime \prime }+{y^{\prime }}^{2}=-1} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 123
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 {-\left (-y \left (x \right )+a \right ) \left (-y \left (x \right )+c_{1} +a \right )}+\frac {c_{1} \arctan \left (\frac {2 y \left (x \right )-2 a -c_{1}}{2 \sqrt {-\left (-y \left (x \right )+a \right ) \left (-y \left (x \right )+c_{1} +a \right )}}\right )}{2}-x -c_{2} &= 0 \\ \sqrt {-\left (-y \left (x \right )+a \right ) \left (-y \left (x \right )+c_{1} +a \right )}-\frac {c_{1} \arctan \left (\frac {2 y \left (x \right )-2 a -c_{1}}{2 \sqrt {-\left (-y \left (x \right )+a \right ) \left (-y \left (x \right )+c_{1} +a \right )}}\right )}{2}-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.945 (sec). Leaf size: 595
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 {\sqrt {2} e^{2 c_1} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}{\sqrt {2} \sqrt {a-\text {$\#$1}}}\right )+2 \sqrt {a-\text {$\#$1}} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}{2 \sqrt {2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {2} e^{2 c_1} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}{\sqrt {2} \sqrt {a-\text {$\#$1}}}\right )+2 \sqrt {a-\text {$\#$1}} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}{2 \sqrt {2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {2} e^{2 (-c_1)} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-2 a+e^{2 (-c_1)}}}{\sqrt {2} \sqrt {a-\text {$\#$1}}}\right )+2 \sqrt {a-\text {$\#$1}} \sqrt {2 \text {$\#$1}-2 a+e^{2 (-c_1)}}}{2 \sqrt {2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {2} e^{2 (-c_1)} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-2 a+e^{2 (-c_1)}}}{\sqrt {2} \sqrt {a-\text {$\#$1}}}\right )+2 \sqrt {a-\text {$\#$1}} \sqrt {2 \text {$\#$1}-2 a+e^{2 (-c_1)}}}{2 \sqrt {2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {2} e^{2 c_1} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}{\sqrt {2} \sqrt {a-\text {$\#$1}}}\right )+2 \sqrt {a-\text {$\#$1}} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}{2 \sqrt {2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {2} e^{2 c_1} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}{\sqrt {2} \sqrt {a-\text {$\#$1}}}\right )+2 \sqrt {a-\text {$\#$1}} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}{2 \sqrt {2}}\&\right ][x+c_2] \\ \end{align*}