ODE
\[ y'(x)^2-2 (1-3 y(x)) y'(x)-(4-9 y(x)) y(x)=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.623927 (sec), leaf count = 4913
\[\left \{\left \{y(x)\to \frac {1}{18} \left (-\sqrt {\frac {9 e^{6 (c_1-2 x)} \left (8 e^{6 x}+e^{6 c_1}\right )}{\sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-9 e^{6 c_1-6 x}+9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+4}-9 \sqrt {\frac {e^{6 (c_1-2 x)} \left (-8 e^{6 x}-e^{6 c_1}\right )}{9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-\frac {2}{9} e^{6 c_1-6 x}-\frac {1}{9} \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}-\frac {16 \left (1+27 e^{6 c_1-6 x}\right )}{81 \sqrt {\frac {9 e^{6 (c_1-2 x)} \left (8 e^{6 x}+e^{6 c_1}\right )}{\sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-9 e^{6 c_1-6 x}+9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+4}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (-\sqrt {\frac {9 e^{6 (c_1-2 x)} \left (8 e^{6 x}+e^{6 c_1}\right )}{\sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-9 e^{6 c_1-6 x}+9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+4}+9 \sqrt {\frac {e^{6 (c_1-2 x)} \left (-8 e^{6 x}-e^{6 c_1}\right )}{9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-\frac {2}{9} e^{6 c_1-6 x}-\frac {1}{9} \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}-\frac {16 \left (1+27 e^{6 c_1-6 x}\right )}{81 \sqrt {\frac {9 e^{6 (c_1-2 x)} \left (8 e^{6 x}+e^{6 c_1}\right )}{\sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-9 e^{6 c_1-6 x}+9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+4}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (\sqrt {\frac {9 e^{6 (c_1-2 x)} \left (8 e^{6 x}+e^{6 c_1}\right )}{\sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-9 e^{6 c_1-6 x}+9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+4}-9 \sqrt {\frac {e^{6 (c_1-2 x)} \left (-8 e^{6 x}-e^{6 c_1}\right )}{9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-\frac {2}{9} e^{6 c_1-6 x}-\frac {1}{9} \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+\frac {16 \left (1+27 e^{6 c_1-6 x}\right )}{81 \sqrt {\frac {9 e^{6 (c_1-2 x)} \left (8 e^{6 x}+e^{6 c_1}\right )}{\sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-9 e^{6 c_1-6 x}+9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+4}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (\sqrt {\frac {9 e^{6 (c_1-2 x)} \left (8 e^{6 x}+e^{6 c_1}\right )}{\sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-9 e^{6 c_1-6 x}+9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+4}+9 \sqrt {\frac {e^{6 (c_1-2 x)} \left (-8 e^{6 x}-e^{6 c_1}\right )}{9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-\frac {2}{9} e^{6 c_1-6 x}-\frac {1}{9} \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+\frac {16 \left (1+27 e^{6 c_1-6 x}\right )}{81 \sqrt {\frac {9 e^{6 (c_1-2 x)} \left (8 e^{6 x}+e^{6 c_1}\right )}{\sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}}-9 e^{6 c_1-6 x}+9 \sqrt [3]{e^{-36 x} \left (8 \sqrt {e^{42 x+12 c_1} \left (e^{6 x}-e^{6 c_1}\right ){}^3}-e^{18 (x+c_1)}+20 e^{12 (2 x+c_1)}+8 e^{6 (5 x+c_1)}\right )}+4}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (-\sqrt {\frac {e^{-12 (x+c_1)} \left (9+72 e^{6 (x+c_1)}\right )}{\sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}-9 e^{-6 (x+c_1)}+9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+4}-9 \sqrt {-\frac {16 \left (1+27 e^{-6 (x+c_1)}\right )}{81 \sqrt {\frac {e^{-12 (x+c_1)} \left (9+72 e^{6 (x+c_1)}\right )}{\sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}-9 e^{-6 (x+c_1)}+9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+4}}-\frac {2}{9} e^{-6 (x+c_1)}-\frac {1}{9} \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+\frac {e^{-12 (x+c_1)} \left (-1-8 e^{6 (x+c_1)}\right )}{9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (-\sqrt {\frac {e^{-12 (x+c_1)} \left (9+72 e^{6 (x+c_1)}\right )}{\sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}-9 e^{-6 (x+c_1)}+9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+4}+9 \sqrt {-\frac {16 \left (1+27 e^{-6 (x+c_1)}\right )}{81 \sqrt {\frac {e^{-12 (x+c_1)} \left (9+72 e^{6 (x+c_1)}\right )}{\sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}-9 e^{-6 (x+c_1)}+9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+4}}-\frac {2}{9} e^{-6 (x+c_1)}-\frac {1}{9} \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+\frac {e^{-12 (x+c_1)} \left (-1-8 e^{6 (x+c_1)}\right )}{9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (\sqrt {\frac {e^{-12 (x+c_1)} \left (9+72 e^{6 (x+c_1)}\right )}{\sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}-9 e^{-6 (x+c_1)}+9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+4}-9 \sqrt {\frac {16 \left (1+27 e^{-6 (x+c_1)}\right )}{81 \sqrt {\frac {e^{-12 (x+c_1)} \left (9+72 e^{6 (x+c_1)}\right )}{\sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}-9 e^{-6 (x+c_1)}+9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+4}}-\frac {2}{9} e^{-6 (x+c_1)}-\frac {1}{9} \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+\frac {e^{-12 (x+c_1)} \left (-1-8 e^{6 (x+c_1)}\right )}{9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (\sqrt {\frac {e^{-12 (x+c_1)} \left (9+72 e^{6 (x+c_1)}\right )}{\sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}-9 e^{-6 (x+c_1)}+9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+4}+9 \sqrt {\frac {16 \left (1+27 e^{-6 (x+c_1)}\right )}{81 \sqrt {\frac {e^{-12 (x+c_1)} \left (9+72 e^{6 (x+c_1)}\right )}{\sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}-9 e^{-6 (x+c_1)}+9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+4}}-\frac {2}{9} e^{-6 (x+c_1)}-\frac {1}{9} \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}+\frac {e^{-12 (x+c_1)} \left (-1-8 e^{6 (x+c_1)}\right )}{9 \sqrt [3]{e^{-36 (x+c_1)} \left (8 \sqrt {e^{42 (x+c_1)} \left (-1+e^{6 (x+c_1)}\right ){}^3}-e^{18 (x+c_1)}+20 e^{24 (x+c_1)}+8 e^{30 (x+c_1)}\right )}}+\frac {8}{81}}+2\right )\right \}\right \}\]
Maple ✓
cpu = 8.926 (sec), leaf count = 123
\[\left [y \left (x \right ) = {\frac {4}{9}}, y \left (x \right ) = \frac {\RootOf \left (\textit {\_Z}^{8} {\mathrm e}^{24 x}+24 \textit {\_Z}^{7} {\mathrm e}^{24 x}+240 \textit {\_Z}^{6} {\mathrm e}^{24 x}+1280 \textit {\_Z}^{5} {\mathrm e}^{24 x}+\left (3840 \,{\mathrm e}^{24 x}-1458 \,{\mathrm e}^{12 x} \textit {\_C1} \right ) \textit {\_Z}^{4}+\left (6144 \,{\mathrm e}^{24 x}+75816 \,{\mathrm e}^{12 x} \textit {\_C1} \right ) \textit {\_Z}^{3}+\left (4096 \,{\mathrm e}^{24 x}-209952 \,{\mathrm e}^{12 x} \textit {\_C1} \right ) \textit {\_Z}^{2}-23328 \textit {\_Z} \,{\mathrm e}^{12 x} \textit {\_C1} -11664 \,{\mathrm e}^{12 x} \textit {\_C1} +531441 \textit {\_C1}^{2}\right )}{9}+\frac {4}{9}\right ]\] Mathematica raw input
DSolve[-((4 - 9*y[x])*y[x]) - 2*(1 - 3*y[x])*y'[x] + y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2 - Sqrt[4 - 9*E^(-6*x + 6*C[1]) + (9*E^(6*(-2*x + C[1]))*(8*E^(6*x) + E^(6*C[1])))/((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3) + 9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)] - 9*Sqrt[8/81 - (2*E^(-6*x + 6*C[1]))/9 + (E^(6*(-2*x + C[1]))*(-8*E^(6*x) - E^(6*C[1])))/(9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)) - ((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)/9 - (16*(1 + 27*E^(-6*x + 6*C[1])))/(81*Sqrt[4 - 9*E^(-6*x + 6*C[1]) + (9*E^(6*(-2*x + C[1]))*(8*E^(6*x) + E^(6*C[1])))/((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3) + 9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)])])/18}, {y[x] -> (2 - Sqrt[4 - 9*E^(-6*x + 6*C[1]) + (9*E^(6*(-2*x + C[1]))*(8*E^(6*x) + E^(6*C[1])))/((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3) + 9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)] + 9*Sqrt[8/81 - (2*E^(-6*x + 6*C[1]))/9 + (E^(6*(-2*x + C[1]))*(-8*E^(6*x) - E^(6*C[1])))/(9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)) - ((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)/9 - (16*(1 + 27*E^(-6*x + 6*C[1])))/(81*Sqrt[4 - 9*E^(-6*x + 6*C[1]) + (9*E^(6*(-2*x + C[1]))*(8*E^(6*x) + E^(6*C[1])))/((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3) + 9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)])])/18}, {y[x] -> (2 + Sqrt[4 - 9*E^(-6*x + 6*C[1]) + (9*E^(6*(-2*x + C[1]))*(8*E^(6*x) + E^(6*C[1])))/((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3) + 9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)] - 9*Sqrt[8/81 - (2*E^(-6*x + 6*C[1]))/9 + (E^(6*(-2*x + C[1]))*(-8*E^(6*x) - E^(6*C[1])))/(9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)) - ((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)/9 + (16*(1 + 27*E^(-6*x + 6*C[1])))/(81*Sqrt[4 - 9*E^(-6*x + 6*C[1]) + (9*E^(6*(-2*x + C[1]))*(8*E^(6*x) + E^(6*C[1])))/((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3) + 9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)])])/18}, {y[x] -> (2 + Sqrt[4 - 9*E^(-6*x + 6*C[1]) + (9*E^(6*(-2*x + C[1]))*(8*E^(6*x) + E^(6*C[1])))/((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3) + 9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)] + 9*Sqrt[8/81 - (2*E^(-6*x + 6*C[1]))/9 + (E^(6*(-2*x + C[1]))*(-8*E^(6*x) - E^(6*C[1])))/(9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)) - ((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)/9 + (16*(1 + 27*E^(-6*x + 6*C[1])))/(81*Sqrt[4 - 9*E^(-6*x + 6*C[1]) + (9*E^(6*(-2*x + C[1]))*(8*E^(6*x) + E^(6*C[1])))/((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3) + 9*((-E^(18*(x + C[1])) + 20*E^(12*(2*x + C[1])) + 8*E^(6*(5*x + C[1])) + 8*Sqrt[E^(42*x + 12*C[1])*(E^(6*x) - E^(6*C[1]))^3])/E^(36*x))^(1/3)])])/18}, {y[x] -> (2 - Sqrt[4 - 9/E^(6*(x + C[1])) + (9 + 72*E^(6*(x + C[1])))/(E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) + 9*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)] - 9*Sqrt[8/81 - 2/(9*E^(6*(x + C[1]))) + (-1 - 8*E^(6*(x + C[1])))/(9*E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) - ((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)/9 - (16*(1 + 27/E^(6*(x + C[1]))))/(81*Sqrt[4 - 9/E^(6*(x + C[1])) + (9 + 72*E^(6*(x + C[1])))/(E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) + 9*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)])])/18}, {y[x] -> (2 - Sqrt[4 - 9/E^(6*(x + C[1])) + (9 + 72*E^(6*(x + C[1])))/(E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) + 9*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)] + 9*Sqrt[8/81 - 2/(9*E^(6*(x + C[1]))) + (-1 - 8*E^(6*(x + C[1])))/(9*E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) - ((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)/9 - (16*(1 + 27/E^(6*(x + C[1]))))/(81*Sqrt[4 - 9/E^(6*(x + C[1])) + (9 + 72*E^(6*(x + C[1])))/(E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) + 9*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)])])/18}, {y[x] -> (2 + Sqrt[4 - 9/E^(6*(x + C[1])) + (9 + 72*E^(6*(x + C[1])))/(E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) + 9*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)] - 9*Sqrt[8/81 - 2/(9*E^(6*(x + C[1]))) + (-1 - 8*E^(6*(x + C[1])))/(9*E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) - ((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)/9 + (16*(1 + 27/E^(6*(x + C[1]))))/(81*Sqrt[4 - 9/E^(6*(x + C[1])) + (9 + 72*E^(6*(x + C[1])))/(E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) + 9*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)])])/18}, {y[x] -> (2 + Sqrt[4 - 9/E^(6*(x + C[1])) + (9 + 72*E^(6*(x + C[1])))/(E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) + 9*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)] + 9*Sqrt[8/81 - 2/(9*E^(6*(x + C[1]))) + (-1 - 8*E^(6*(x + C[1])))/(9*E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) - ((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)/9 + (16*(1 + 27/E^(6*(x + C[1]))))/(81*Sqrt[4 - 9/E^(6*(x + C[1])) + (9 + 72*E^(6*(x + C[1])))/(E^(12*(x + C[1]))*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)) + 9*((-E^(18*(x + C[1])) + 20*E^(24*(x + C[1])) + 8*E^(30*(x + C[1])) + 8*Sqrt[E^(42*(x + C[1]))*(-1 + E^(6*(x + C[1])))^3])/E^(36*(x + C[1])))^(1/3)])])/18}}
Maple raw input
dsolve(diff(y(x),x)^2-2*(1-3*y(x))*diff(y(x),x)-(4-9*y(x))*y(x) = 0, y(x))
Maple raw output
[y(x) = 4/9, y(x) = 1/9*RootOf(_Z^8*exp(x)^24+24*_Z^7*exp(x)^24+240*_Z^6*exp(x)^24+1280*_Z^5*exp(x)^24+(3840*exp(x)^24-1458*exp(x)^12*_C1)*_Z^4+(6144*exp(x)^24+75816*exp(x)^12*_C1)*_Z^3+(4096*exp(x)^24-209952*exp(x)^12*_C1)*_Z^2-23328*_Z*exp(x)^12*_C1-11664*exp(x)^12*_C1+531441*_C1^2)+4/9]