Internal problem ID [3570]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 839.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class C], _dAlembert]
Solve \begin {gather*} \boxed {4 \left (y^{\prime }\right )^{2}+2 \,{\mathrm e}^{2 x -2 y} y^{\prime }-{\mathrm e}^{2 x -2 y}=0} \end {gather*}
✓ Solution by Maple
Time used: 1.622 (sec). Leaf size: 137
dsolve(4*diff(y(x),x)^2+2*exp(2*x-2*y(x))*diff(y(x),x)-exp(2*x-2*y(x)) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1}-\arctanh \left (\frac {1}{\RootOf \left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\RootOf \left (16 \,{\mathrm e}^{2 \textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +c_{1}\right )\right )+8 \,{\mathrm e}^{\textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +c_{1}\right )\right )+\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +c_{1}\right )-4 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right ) \\ y \relax (x ) = c_{1}+\arctanh \left (\frac {1}{\RootOf \left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\RootOf \left (16 \,{\mathrm e}^{2 \textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +c_{1}\right )\right )+8 \,{\mathrm e}^{\textit {\_Z}} \left (\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +c_{1}\right )\right )+\tanh ^{2}\left (-\frac {\textit {\_Z}}{2}-x +c_{1}\right )-4 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.013 (sec). Leaf size: 176
DSolve[4 (y'[x])^2+2 Exp[2 x-2 y[x]] y'[x]-Exp[2 x-2 y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [y(x)-\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \tanh ^{-1}\left (\frac {e^x}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}=c_1,y(x)\right ] \\ \text {Solve}\left [y(x)+\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \tanh ^{-1}\left (\frac {e^x}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}=c_1,y(x)\right ] \\ y(x)\to \frac {1}{2} \left (\log \left (-\frac {e^{4 x}}{4}\right )-2 x\right ) \\ \end{align*}