Internal problem ID [8794]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 459.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {{\mathrm e}^{-2 x} {y^{\prime }}^{2}-\left (y^{\prime }-1\right )^{2}+{\mathrm e}^{-2 y}=0} \]
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 128
dsolve(exp(-2*x)*diff(y(x),x)^2-(diff(y(x),x)-1)^2+exp(-2*y(x)) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} -\ln \left (\frac {-\sqrt {{\mathrm e}^{-2 x +4 c_{1}}-{\mathrm e}^{-2 x +2 c_{1}}}\, {\mathrm e}^{2 x}-{\mathrm e}^{2 c_{1}}}{-{\mathrm e}^{2 c_{1} +2 x}+{\mathrm e}^{2 c_{1}}+{\mathrm e}^{2 x}}\right ) \\ y \left (x \right ) &= c_{1} -\ln \left (\frac {\sqrt {{\mathrm e}^{-2 x +4 c_{1}}-{\mathrm e}^{-2 x +2 c_{1}}}\, {\mathrm e}^{2 x}-{\mathrm e}^{2 c_{1}}}{-{\mathrm e}^{2 c_{1} +2 x}+{\mathrm e}^{2 c_{1}}+{\mathrm e}^{2 x}}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 24.762 (sec). Leaf size: 583
DSolve[E^(-2*y[x]) - (-1 + y'[x])^2 + y'[x]^2/E^(2*x)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {\left (e^{2 \text {arctanh}\left (1-2 e^x\right )+x}+e^x-1\right ) \sqrt {e^{2 y(x)}+e^{2 x}-1} e^{y(x)-2 \text {arctanh}\left (1-2 e^x\right )} \log \left (\sqrt {e^{2 y(x)}+e^{2 x}-1}+e^{y(x)}\right )}{\sqrt {e^{2 (y(x)+x)} \left (e^{2 y(x)}+e^{2 x}-1\right )}}-y(x)+\log \left (e^{y(x)}\right )-\frac {1}{2} \log \left (e^{y(x)}-1\right )-\frac {1}{2} \log \left (e^{y(x)}+1\right )+\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x-e^{2 x}\right )+\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x+e^{2 x}\right )-x-\frac {1}{2} \log \left (1-e^x\right )-\frac {1}{2} \log \left (e^x-1\right )&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\left (e^{2 \text {arctanh}\left (1-2 e^x\right )+x}+e^x-1\right ) \sqrt {e^{2 y(x)}+e^{2 x}-1} e^{y(x)-2 \text {arctanh}\left (1-2 e^x\right )} \log \left (\sqrt {e^{2 y(x)}+e^{2 x}-1}+e^{y(x)}\right )}{\sqrt {e^{2 (y(x)+x)} \left (e^{2 y(x)}+e^{2 x}-1\right )}}-\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x-e^{2 x}\right )-\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x+e^{2 x}\right )+\frac {1}{2} \left (2 y(x)-2 \log \left (e^{y(x)}\right )+\log \left (e^{y(x)}-1\right )+\log \left (e^{y(x)}+1\right )\right )+x-\frac {1}{2} \log \left (1-e^x\right )+\frac {1}{2} \log \left (e^x-1\right )+\log \left (e^x+1\right )&=c_1,y(x)\right ] \\ y(x)\to \log \left (-\sqrt {1-e^{2 x}}\right ) \\ y(x)\to \frac {1}{2} \log \left (1-e^{2 x}\right ) \\ \end{align*}