Internal problem ID [2294]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 35, page 157
Problem number: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2}=-1} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 59
dsolve(y(x)*diff(y(x),x$2)+1=diff(y(x),x)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ y \left (x \right ) &= \frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.25 (sec). Leaf size: 85
DSolve[y[x]*y''[x]+1==y'[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i e^{-c_1} \tanh \left (e^{c_1} (x+c_2)\right )}{\sqrt {-\text {sech}^2\left (e^{c_1} (x+c_2)\right )}} \\ y(x)\to \frac {i e^{-c_1} \tanh \left (e^{c_1} (x+c_2)\right )}{\sqrt {-\text {sech}^2\left (e^{c_1} (x+c_2)\right )}} \\ \end{align*}