29.29.16 problem 838

Internal problem ID [5421]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 838
Date solved : Monday, January 27, 2025 at 11:21:44 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} 4 {y^{\prime }}^{2}+2 x \,{\mathrm e}^{-2 y} y^{\prime }-{\mathrm e}^{-2 y}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.183 (sec). Leaf size: 103

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 ) &= -\ln \left (2\right )-\frac {\ln \left (-\frac {1}{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 (4 \,{\mathrm e}^{\textit {\_Z}} \sinh \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 (4 \,{\mathrm e}^{\textit {\_Z}} \sinh \left (-\frac {\textit {\_Z}}{2}+c_{1} \right )^{2}-x^{2}\right )}\right )}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.367 (sec). Leaf size: 76

DSolve[4 (D[y[x],x])^2+2 x Exp[-2 y[x]] D[y[x],x]-Exp[-2 y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} \log \left (\frac {1}{4} \left (e^{2 c_1}-2 e^{c_1} x\right )\right ) \\ \text {Solve}\left [2 y(x)-\log \left (\sqrt {x^2+4 e^{2 y(x)}}+x\right )&=c_1,y(x)\right ] \\ y(x)\to \frac {1}{2} \log \left (-\frac {x^2}{4}\right ) \\ \end{align*}