Internal problem ID [4069]
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]
\[ \boxed {{y^{\prime }}^{2}-{\mathrm e}^{4 x -2 y} \left (y^{\prime }-1\right )=0} \]
✓ Solution by Maple
Time used: 1.969 (sec). Leaf size: 307
dsolve(diff(y(x),x)^2 = exp(4*x-2*y(x))*(diff(y(x),x)-1),y(x), singsol=all)
\begin{align*} \frac {-\frac {\sqrt {-4 \,{\mathrm e}^{-4 y \left (x \right )+8 x} {\mathrm e}^{-4 x +2 y \left (x \right )}+{\mathrm e}^{-4 y \left (x \right )+8 x}}\, {\mathrm e}^{-4 x +2 y \left (x \right )} \operatorname {arctanh}\left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}\right )}{2}+\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}\, \left (x -\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y \left (x \right )}+1\right )}{4}-c_{1} +\frac {\ln \left ({\mathrm e}^{-2 x +y \left (x \right )}\right )}{2}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y \left (x \right )}-1\right )}{4}\right )}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}} &= 0 \\ \frac {\frac {\sqrt {-4 \,{\mathrm e}^{-4 y \left (x \right )+8 x} {\mathrm e}^{-4 x +2 y \left (x \right )}+{\mathrm e}^{-4 y \left (x \right )+8 x}}\, {\mathrm e}^{-4 x +2 y \left (x \right )} \operatorname {arctanh}\left (\frac {1}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}}\right )}{2}+\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}\, \left (x -\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y \left (x \right )}+1\right )}{4}-c_{1} +\frac {\ln \left ({\mathrm e}^{-2 x +y \left (x \right )}\right )}{2}+\frac {\ln \left (4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}-1\right )}{4}-\frac {\ln \left (2 \,{\mathrm e}^{-2 x +y \left (x \right )}-1\right )}{4}\right )}{\sqrt {-4 \,{\mathrm e}^{-4 x +2 y \left (x \right )}+1}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.551 (sec). Leaf size: 383
DSolve[(y'[x])^2==Exp[4 x -2 y[x]] (y'[x]-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {e^{-2 x} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}} \text {arctanh}\left (\frac {-\sqrt {e^{4 x}-4 e^{2 y(x)}}+e^{2 x}+1}{\sqrt {e^{4 x}-4 e^{2 y(x)}}-e^{2 x}+1}\right )}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}-\frac {e^{-2 x} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}} y(x)}{2 \sqrt {e^{4 x}-4 e^{2 y(x)}}}+\frac {y(x)}{2}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {e^{-2 x} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}} \text {arctanh}\left (\frac {-\sqrt {e^{4 x}-4 e^{2 y(x)}}+e^{2 x}+1}{\sqrt {e^{4 x}-4 e^{2 y(x)}}-e^{2 x}+1}\right )}{\sqrt {e^{4 x}-4 e^{2 y(x)}}}+\frac {\left (\sqrt {e^{4 x}-4 e^{2 y(x)}} \sqrt {e^{8 x}-4 e^{2 y(x)+4 x}}-4 e^{2 (y(x)+x)}+e^{6 x}\right ) y(x)}{2 e^{6 x}-8 e^{2 (y(x)+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*}