Internal problem ID [10300]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 58.
Independent variable absent. Page 135
Problem number: Ex 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2}+1=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 79
dsolve(y(x)*diff(y(x),x$2)-diff(y(x),x)^2+1=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {c_{1} \left ({\mathrm e}^{-\frac {2 x}{c_{1}}} {\mathrm e}^{-\frac {2 c_{2}}{c_{1}}}-1\right ) {\mathrm e}^{\frac {x}{c_{1}}} {\mathrm e}^{\frac {c_{2}}{c_{1}}}}{2} \\ y \left (x \right ) = \frac {c_{1} \left ({\mathrm e}^{\frac {2 x}{c_{1}}} {\mathrm e}^{\frac {2 c_{2}}{c_{1}}}-1\right ) {\mathrm e}^{-\frac {x}{c_{1}}} {\mathrm e}^{-\frac {c_{2}}{c_{1}}}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.145 (sec). Leaf size: 85
DSolve[y[x]*y''[x]-y'[x]^2+1==0,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*}