Internal problem ID [3569]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 838.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 \left (y^{\prime }\right )^{2}+2 x \,{\mathrm e}^{-2 y} y^{\prime }-{\mathrm e}^{-2 y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.148 (sec). Leaf size: 122
dsolve(4*diff(y(x),x)^2+2*x*exp(-2*y(x))*diff(y(x),x)-exp(-2*y(x)) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\ln \left (-\frac {4}{x^{2}}\right )}{2} \\ y \relax (x ) = c_{1}-\arctanh \left (\frac {x}{\RootOf \left (\textit {\_Z}^{2}-x^{2}-4 \,{\mathrm e}^{\RootOf \left (x^{2} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+c_{1}\right )\right )+4 \,{\mathrm e}^{\textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+c_{1}\right )\right )-x^{2}\right )}\right )}\right ) \\ y \relax (x ) = c_{1}+\arctanh \left (\frac {x}{\RootOf \left (\textit {\_Z}^{2}-x^{2}-4 \,{\mathrm e}^{\RootOf \left (x^{2} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+c_{1}\right )\right )+4 \,{\mathrm e}^{\textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}+c_{1}\right )\right )-x^{2}\right )}\right )}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.664 (sec). Leaf size: 119
DSolve[4 (y'[x])^2+2 x Exp[-2 y[x]] y'[x]-Exp[-2 y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \log \left (-e^{\frac {c_1}{2}} \sqrt {-x+e^{c_1}}\right ) \\ y(x)\to \log \left (e^{\frac {c_1}{2}} \sqrt {-x+e^{c_1}}\right ) \\ y(x)\to \log \left (-e^{\frac {c_1}{2}} \sqrt {x+e^{c_1}}\right ) \\ y(x)\to \log \left (e^{\frac {c_1}{2}} \sqrt {x+e^{c_1}}\right ) \\ y(x)\to \frac {1}{2} \log \left (-\frac {x^2}{4}\right ) \\ \end{align*}