Internal problem ID [3561]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 830.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class C], _dAlembert]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-{\mathrm e}^{4 x -2 y} \left (y^{\prime }-1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.954 (sec). Leaf size: 259
dsolve(diff(y(x),x)^2 = exp(4*x-2*y(x))*(diff(y(x),x)-1),y(x), singsol=all)
\begin{align*} x -\frac {\sqrt {-\left (4 \,{\mathrm e}^{-4 x +2 y \relax (x )}-1\right ) {\mathrm e}^{-4 y \relax (x )+8 x}}\, {\mathrm e}^{-4 x +2 y \relax (x )} \arctanh \left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \relax (x )}+1}}\right )}{2 \sqrt {-4 \,{\mathrm e}^{-4 x +2 y \relax (x )}+1}}-\frac {\ln \left (2 \,{\mathrm e}^{y \relax (x )-2 x}-1\right )}{4}+\frac {\ln \left ({\mathrm e}^{y \relax (x )-2 x}\right )}{2}-\frac {\ln \left (2 \,{\mathrm e}^{y \relax (x )-2 x}+1\right )}{4}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \relax (x )}-1\right )}{4}-c_{1} = 0 \\ x -\frac {\ln \left (2 \,{\mathrm e}^{y \relax (x )-2 x}-1\right )}{4}+\frac {\ln \left ({\mathrm e}^{y \relax (x )-2 x}\right )}{2}-\frac {\ln \left (2 \,{\mathrm e}^{y \relax (x )-2 x}+1\right )}{4}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \relax (x )}-1\right )}{4}+\frac {\sqrt {-\left (4 \,{\mathrm e}^{-4 x +2 y \relax (x )}-1\right ) {\mathrm e}^{-4 y \relax (x )+8 x}}\, {\mathrm e}^{-4 x +2 y \relax (x )} \arctanh \left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \relax (x )}+1}}\right )}{2 \sqrt {-4 \,{\mathrm e}^{-4 x +2 y \relax (x )}+1}}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.14 (sec). Leaf size: 197
DSolve[(y'[x])^2==Exp[4 x -2 y[x]] (y'[x]-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {y(x)}{2}-\frac {e^{-2 x} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}} \tanh ^{-1}\left (\frac {e^{2 x}}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}\right )}{2 \sqrt {e^{4 x}-4 e^{2 y(x)}}}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {y(x)}{2}+\frac {e^{-2 x} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}} \tanh ^{-1}\left (\frac {e^{2 x}}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}\right )}{2 \sqrt {e^{4 x}-4 e^{2 y(x)}}}=c_1,y(x)\right ] \\ y(x)\to \frac {1}{2} \left (\log \left (\frac {e^{8 x}}{4}\right )-4 x\right ) \\ \end{align*}