Internal problem ID [8553]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 973.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Abel]
Solve \begin {gather*} \boxed {y^{\prime }-y \left (y^{2}+y \,{\mathrm e}^{b x}+{\mathrm e}^{2 b x}\right ) {\mathrm e}^{-2 b x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 135
dsolve(diff(y(x),x) = y(x)*(y(x)^2+y(x)*exp(b*x)+exp(b*x)^2)/exp(b*x)^2,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\tan \left (\RootOf \left (2 \textit {\_Z} \,{\mathrm e}^{b x}-\sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}\, \ln \left (\frac {4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) b -3 \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 b -3}{\left (\tan \left (\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}+{\mathrm e}^{b x}\right )^{2}}\right )+c_{1} \sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}-2 x \sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}\right )\right ) \sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}}{2}-\frac {{\mathrm e}^{b x}}{2} \]
✓ Solution by Mathematica
Time used: 0.268 (sec). Leaf size: 146
DSolve[y'[x] == (y[x]*(E^(2*b*x) + E^(b*x)*y[x] + y[x]^2))/E^(2*b*x),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {1}{3} (9 b-7)^{2/3} \text {RootSum}\left [\text {$\#$1}^3 (9 b-7)^{2/3}-9 \text {$\#$1} b+6 \text {$\#$1}+(9 b-7)^{2/3}\&,\frac {\log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b-7) e^{-3 b x}}}-\text {$\#$1}\right )}{\text {$\#$1}^2 \left (-(9 b-7)^{2/3}\right )+3 b-2}\&\right ]=\frac {1}{9} x e^{2 b x} \left ((9 b-7) e^{-3 b x}\right )^{2/3}+c_1,y(x)\right ] \]