Internal problem ID [8551]
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]
\[ \boxed {y^{\prime }-y \left (y^{2}+y \,{\mathrm e}^{b x}+{\mathrm e}^{2 b x}\right ) {\mathrm e}^{-2 b x}=0} \]
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 136
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)
\[ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\ln \left (-\frac {4 \tan \left (\textit {\_Z} \right )^{2} b -3 \tan \left (\textit {\_Z} \right )^{2}+4 b -3}{\left (\tan \left (\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{2 x b} \left (-3+4 b \right )}-{\mathrm e}^{x b}\right )^{2}}\right ) \sqrt {-{\mathrm e}^{2 x b} \left (-3+4 b \right )}+2 \textit {\_Z} \,{\mathrm e}^{x b}+\sqrt {-{\mathrm e}^{2 x b} \left (-3+4 b \right )}\, c_{1} +2 \sqrt {-{\mathrm e}^{2 x b} \left (-3+4 b \right )}\, x \right )\right ) \sqrt {-{\mathrm e}^{2 x b} \left (-3+4 b \right )}}{2}-\frac {{\mathrm e}^{x b}}{2} \]
✓ Solution by Mathematica
Time used: 1.043 (sec). Leaf size: 1225
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 {-2 \sqrt {3} \sqrt [3]{7-9 b} \left (\sqrt [3]{7-9 b}+\sqrt [3]{9 b-7}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{7-9 b} \left (3 e^{-2 b x} y(x)+e^{-b x}\right )}{\sqrt [3]{(9 b-7) e^{-3 b x}}}-1}{\sqrt {3}}\right )-2 \sqrt [3]{7-9 b} \left (\sqrt [3]{7-9 b}-\sqrt [3]{9 b-7}\right ) \log \left (\frac {\sqrt [3]{7-9 b} \left (3 e^{-2 b x} y(x)+e^{-b x}\right )}{\sqrt [3]{(9 b-7) e^{-3 b x}}}+1\right )+\sqrt [3]{7-9 b} \left (\sqrt [3]{7-9 b}-\sqrt [3]{9 b-7}\right ) \log \left (\frac {(7-9 b)^{2/3} \left (3 e^{-2 b x} y(x)+e^{-b x}\right )^2}{\left ((9 b-7) e^{-3 b x}\right )^{2/3}}-\frac {\sqrt [3]{7-9 b} \left (3 e^{-2 b x} y(x)+e^{-b x}\right )}{\sqrt [3]{(9 b-7) e^{-3 b x}}}+1\right )-2 (9 b-7)^{2/3} \log \left (\frac {(7-9 b) e^{3 b x} \left (3 e^{-2 b x} y(x)+e^{-b x}\right )^3}{9 b-7}+1\right )+2 \text {RootSum}\left [-9 b \text {$\#$1}^6+7 \text {$\#$1}^6-27 b \text {$\#$1}^3+20 \text {$\#$1}^3+81 b^2-126 b+49\&,\frac {9 b (9 b-7)^{2/3} \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}^5-7 (9 b-7)^{2/3} \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}^5+9 b \sqrt [3]{9 b-7} \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}^4-7 \sqrt [3]{9 b-7} \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}^4+81 b^2 \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}^3-135 b \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}^3+56 \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}^3+27 b (9 b-7)^{2/3} \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}^2-19 (9 b-7)^{2/3} \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+81 b^2 \sqrt [3]{9 b-7} \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}-108 b \sqrt [3]{9 b-7} \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}+35 \sqrt [3]{9 b-7} \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}+162 b^2 \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 )-252 b \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 )+98 \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 )}{18 b \text {$\#$1}^5-14 \text {$\#$1}^5+27 b \text {$\#$1}^2-20 \text {$\#$1}^2}\&\right ]}{54 (b-1)}=\frac {1}{9} e^{2 b x} \left ((9 b-7) e^{-3 b x}\right )^{2/3} x+c_1,y(x)\right ] \]