Internal problem ID [9028]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 693.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Abel]
\[ \boxed {y^{\prime }-\left (1+y^{2} {\mathrm e}^{-2 b x}+{\mathrm e}^{-3 b x} y^{3}\right ) {\mathrm e}^{b x}=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 33
dsolve(diff(y(x),x) = (1+y(x)^2*exp(-2*b*x)+y(x)^3*exp(-3*b*x))*exp(b*x),y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{3}+\textit {\_a}^{2}-\textit {\_a} b +1}d \textit {\_a} +c_{1} \right ) {\mathrm e}^{b x} \]
✓ Solution by Mathematica
Time used: 2.849 (sec). Leaf size: 1121
DSolve[y'[x] == E^(b*x)*(1 + y[x]^2/E^(2*b*x) + y[x]^3/E^(3*b*x)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{9} \text {RootSum}\left [-81 b^2 \text {$\#$1}^9-522 b \text {$\#$1}^9-841 \text {$\#$1}^9-243 b^2 \text {$\#$1}^6-1566 b \text {$\#$1}^6-2523 \text {$\#$1}^6+729 b^3 \text {$\#$1}^3+486 b^2 \text {$\#$1}^3-1323 b \text {$\#$1}^3-2496 \text {$\#$1}^3-81 b^2-522 b-841\&,\frac {81 b^2 \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^6+522 b \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^6+841 \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^6+81 b^2 \sqrt [3]{9 b+29} \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^4+288 b \sqrt [3]{9 b+29} \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^4+87 \sqrt [3]{9 b+29} \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^4+162 b^2 \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^3+1044 b \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^3+1682 \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^3+81 b^2 (9 b+29)^{2/3} \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^2+54 b (9 b+29)^{2/3} \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^2+9 (9 b+29)^{2/3} \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}^2+81 b^2 \sqrt [3]{9 b+29} \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}+288 b \sqrt [3]{9 b+29} \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}+87 \sqrt [3]{9 b+29} \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right ) \text {$\#$1}+81 b^2 \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right )+522 b \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right )+841 \log \left (\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b+29) e^{-3 b x}}}-\text {$\#$1}\right )}{81 b^2 \text {$\#$1}^8+522 b \text {$\#$1}^8+841 \text {$\#$1}^8+162 b^2 \text {$\#$1}^5+1044 b \text {$\#$1}^5+1682 \text {$\#$1}^5-243 b^3 \text {$\#$1}^2-162 b^2 \text {$\#$1}^2+441 b \text {$\#$1}^2+832 \text {$\#$1}^2}\&\right ]=\frac {1}{9} e^{2 b x} \left ((9 b+29) e^{-3 b x}\right )^{2/3} x+c_1,y(x)\right ] \]