29.16 problem 838

Internal problem ID [3568]

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]]

\[ \boxed {4 {y^{\prime }}^{2}+2 x \,{\mathrm e}^{-2 y} y^{\prime }-{\mathrm e}^{-2 y}=0} \]

Solution by Maple

Time used: 0.093 (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 \left (x \right ) = -\frac {\ln \left (-\frac {4}{x^{2}}\right )}{2} \\ y \left (x \right ) = c_{1} -\operatorname {arctanh}\left (\frac {x}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-x^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (x^{2} \tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} \right )^{2}+4 \,{\mathrm e}^{\textit {\_Z}} \tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} \right )^{2}-x^{2}\right )}\right )}\right ) \\ y \left (x \right ) = c_{1} +\operatorname {arctanh}\left (\frac {x}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-x^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (x^{2} \tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} \right )^{2}+4 \,{\mathrm e}^{\textit {\_Z}} \tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} \right )^{2}-x^{2}\right )}\right )}\right ) \\ \end{align*}

Solution by Mathematica

Time used: 10.223 (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*}