Internal problem ID [3569]
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]
\[ \boxed {4 {y^{\prime }}^{2}+2 \,{\mathrm e}^{2 x -2 y} y^{\prime }-{\mathrm e}^{2 x -2 y}=0} \]
✓ Solution by Maple
Time used: 1.109 (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 \left (x \right ) = c_{1} -\operatorname {arctanh}\left (\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (16 \tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} -x \right )^{2} {\mathrm e}^{2 \textit {\_Z}}+8 \tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} -x \right )^{2} {\mathrm e}^{\textit {\_Z}}+\tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} -x \right )^{2}-4 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right ) \\ y \left (x \right ) = c_{1} +\operatorname {arctanh}\left (\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (16 \tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} -x \right )^{2} {\mathrm e}^{2 \textit {\_Z}}+8 \tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} -x \right )^{2} {\mathrm e}^{\textit {\_Z}}+\tanh \left (-\frac {\textit {\_Z}}{2}+c_{1} -x \right )^{2}-4 \,{\mathrm e}^{\textit {\_Z}}-1\right )}-1\right )}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.613 (sec). Leaf size: 332
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 [-\frac {2 e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \text {arctanh}\left (\frac {-\sqrt {4 e^{2 y(x)}+e^{2 x}}+e^x+1}{\sqrt {4 e^{2 y(x)}+e^{2 x}}-e^x+1}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}-\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} y(x)}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}+y(x)=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} \text {arctanh}\left (\frac {-\sqrt {4 e^{2 y(x)}+e^{2 x}}+e^x+1}{\sqrt {4 e^{2 y(x)}+e^{2 x}}-e^x+1}\right )}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}+\frac {e^{-x} \sqrt {4 e^{2 (y(x)+x)}+e^{4 x}} y(x)}{\sqrt {4 e^{2 y(x)}+e^{2 x}}}+y(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*}