Internal problem ID [7967]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 387.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+\left (y^{\prime }-y\right ) {\mathrm e}^{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.406 (sec). Leaf size: 118
dsolve(diff(y(x),x)^2+(diff(y(x),x)-y(x))*exp(x) = 0,y(x), singsol=all)
\begin{align*} 2 \ln \left (y \relax (x )\right )+\frac {\sqrt {{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} y \relax (x )}}{y \relax (x )}+4 \arctanh \left (\sqrt {{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} y \relax (x )}\, {\mathrm e}^{-x}\right )-\frac {{\mathrm e}^{x}}{y \relax (x )}-c_{1} = 0 \\ 2 \ln \left (y \relax (x )\right )-\frac {\sqrt {{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} y \relax (x )}}{y \relax (x )}-4 \arctanh \left (\sqrt {{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} y \relax (x )}\, {\mathrm e}^{-x}\right )-\frac {{\mathrm e}^{x}}{y \relax (x )}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.537 (sec). Leaf size: 143
DSolve[y'[x]^2 + E^x*(-y[x] + y'[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {e^{x/2} \sqrt {4 y(x)+e^x}-4 y(x) \log \left (\sqrt {4 y(x)+e^x}-e^{x/2}\right )+e^x}{2 y(x)}=c_1,y(x)\right ] \\ \text {Solve}\left [2 \log (y(x))-\frac {-e^{x/2} \sqrt {4 y(x)+e^x}+4 y(x) \log \left (\sqrt {4 y(x)+e^x}-e^{x/2}\right )+e^x}{2 y(x)}=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}