Internal problem ID [10305]
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 3.
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 y^{\prime \prime }-{\mathrm e}^{y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 22
dsolve(2*diff(y(x),x$2)=exp(y(x)),y(x), singsol=all)
\[ y \relax (x ) = \ln \left (\frac {\tan ^{2}\left (\frac {c_{2}+x}{2 c_{1}}\right )+1}{c_{1}^{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.059 (sec). Leaf size: 27
DSolve[2*y''[x]==Exp[y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \log \left (-c_1 \text {sech}^2\left (\frac {1}{2} \sqrt {c_1} (x+c_2)\right )\right ) \\ \end{align*}