4.17.19 \(y'(x)^2-2 (1-3 y(x)) y'(x)-(4-9 y(x)) y(x)=0\)

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 = 0.571143 (sec), leaf count = 4913

\[\left \{\left \{y(x)\to \frac {1}{18} \left (-\sqrt {\frac {9 e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+4}-9 \sqrt {\frac {e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}-\frac {16 \left (1+27 e^{6 c_1-6 x}\right )}{81 \sqrt {\frac {9 e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+4}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (-\sqrt {\frac {9 e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+4}+9 \sqrt {\frac {e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}-\frac {16 \left (1+27 e^{6 c_1-6 x}\right )}{81 \sqrt {\frac {9 e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+4}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (\sqrt {\frac {9 e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+4}-9 \sqrt {\frac {e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+\frac {16 \left (1+27 e^{6 c_1-6 x}\right )}{81 \sqrt {\frac {9 e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+4}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (\sqrt {\frac {9 e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+4}+9 \sqrt {\frac {e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+\frac {16 \left (1+27 e^{6 c_1-6 x}\right )}{81 \sqrt {\frac {9 e^{6 \left (c_1-2 x\right )} \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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\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 \left (x+c_1\right )}+20 e^{12 \left (2 x+c_1\right )}+8 e^{6 \left (5 x+c_1\right )}\right )}+4}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (-\sqrt {\frac {e^{-12 \left (x+c_1\right )} \left (9+72 e^{6 \left (x+c_1\right )}\right )}{\sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}-9 e^{-6 \left (x+c_1\right )}+9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+4}-9 \sqrt {-\frac {16 \left (1+27 e^{-6 \left (x+c_1\right )}\right )}{81 \sqrt {\frac {e^{-12 \left (x+c_1\right )} \left (9+72 e^{6 \left (x+c_1\right )}\right )}{\sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}-9 e^{-6 \left (x+c_1\right )}+9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+4}}-\frac {2}{9} e^{-6 \left (x+c_1\right )}-\frac {1}{9} \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+\frac {e^{-12 \left (x+c_1\right )} \left (-1-8 e^{6 \left (x+c_1\right )}\right )}{9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (-\sqrt {\frac {e^{-12 \left (x+c_1\right )} \left (9+72 e^{6 \left (x+c_1\right )}\right )}{\sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}-9 e^{-6 \left (x+c_1\right )}+9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+4}+9 \sqrt {-\frac {16 \left (1+27 e^{-6 \left (x+c_1\right )}\right )}{81 \sqrt {\frac {e^{-12 \left (x+c_1\right )} \left (9+72 e^{6 \left (x+c_1\right )}\right )}{\sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}-9 e^{-6 \left (x+c_1\right )}+9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+4}}-\frac {2}{9} e^{-6 \left (x+c_1\right )}-\frac {1}{9} \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+\frac {e^{-12 \left (x+c_1\right )} \left (-1-8 e^{6 \left (x+c_1\right )}\right )}{9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (\sqrt {\frac {e^{-12 \left (x+c_1\right )} \left (9+72 e^{6 \left (x+c_1\right )}\right )}{\sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}-9 e^{-6 \left (x+c_1\right )}+9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+4}-9 \sqrt {\frac {16 \left (1+27 e^{-6 \left (x+c_1\right )}\right )}{81 \sqrt {\frac {e^{-12 \left (x+c_1\right )} \left (9+72 e^{6 \left (x+c_1\right )}\right )}{\sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}-9 e^{-6 \left (x+c_1\right )}+9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+4}}-\frac {2}{9} e^{-6 \left (x+c_1\right )}-\frac {1}{9} \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+\frac {e^{-12 \left (x+c_1\right )} \left (-1-8 e^{6 \left (x+c_1\right )}\right )}{9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}+\frac {8}{81}}+2\right )\right \},\left \{y(x)\to \frac {1}{18} \left (\sqrt {\frac {e^{-12 \left (x+c_1\right )} \left (9+72 e^{6 \left (x+c_1\right )}\right )}{\sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}-9 e^{-6 \left (x+c_1\right )}+9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+4}+9 \sqrt {\frac {16 \left (1+27 e^{-6 \left (x+c_1\right )}\right )}{81 \sqrt {\frac {e^{-12 \left (x+c_1\right )} \left (9+72 e^{6 \left (x+c_1\right )}\right )}{\sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}-9 e^{-6 \left (x+c_1\right )}+9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+4}}-\frac {2}{9} e^{-6 \left (x+c_1\right )}-\frac {1}{9} \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}+\frac {e^{-12 \left (x+c_1\right )} \left (-1-8 e^{6 \left (x+c_1\right )}\right )}{9 \sqrt [3]{e^{-36 \left (x+c_1\right )} \left (8 \sqrt {e^{42 \left (x+c_1\right )} \left (-1+e^{6 \left (x+c_1\right )}\right ){}^3}-e^{18 \left (x+c_1\right )}+20 e^{24 \left (x+c_1\right )}+8 e^{30 \left (x+c_1\right )}\right )}}+\frac {8}{81}}+2\right )\right \}\right \}\]

Maple
cpu = 0.063 (sec), leaf count = 155

\[ \left \{ x+{\frac {\ln \left (9\,y \relax (x ) -4 \right ) }{12}}+{\frac {\ln \left (y \relax (x ) \right ) }{4}}-{\frac {1}{4}\ln \left (\sqrt {1-2\,y \relax (x ) }-1 \right ) }+{\frac {1}{4}\ln \left (\sqrt {1-2\,y \relax (x ) }+1 \right ) }-{\frac {1}{12}\ln \left (3\,\sqrt {1-2\,y \relax (x ) }+1 \right ) }+{\frac {1}{12}\ln \left (3\,\sqrt {1-2\,y \relax (x ) }-1 \right ) }-{\it \_C1}=0,x+{\frac {\ln \left (9\,y \relax (x ) -4 \right ) }{12}}+{\frac {\ln \left (y \relax (x ) \right ) }{4}}+{\frac {1}{4}\ln \left (\sqrt {1-2\,y \relax (x ) }-1 \right ) }-{\frac {1}{4}\ln \left (\sqrt {1-2\,y \relax (x ) }+1 \right ) }+{\frac {1}{12}\ln \left (3\,\sqrt {1-2\,y \relax (x ) }+1 \right ) }-{\frac {1}{12}\ln \left (3\,\sqrt {1-2\,y \relax (x ) }-1 \right ) }-{\it \_C1}=0 \right \} \] 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^(1
8*(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])) + 2
0*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 + S
qrt[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^(3
0*(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^(3
0*(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*Sqr
t[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*Sq
rt[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*Sq
rt[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*Sqr
t[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),'implicit')

Maple raw output

x+1/12*ln(9*y(x)-4)+1/4*ln(y(x))+1/4*ln((1-2*y(x))^(1/2)-1)-1/4*ln((1-2*y(x))^(1
/2)+1)+1/12*ln(3*(1-2*y(x))^(1/2)+1)-1/12*ln(3*(1-2*y(x))^(1/2)-1)-_C1 = 0, x+1/
12*ln(9*y(x)-4)+1/4*ln(y(x))-1/4*ln((1-2*y(x))^(1/2)-1)+1/4*ln((1-2*y(x))^(1/2)+
1)-1/12*ln(3*(1-2*y(x))^(1/2)+1)+1/12*ln(3*(1-2*y(x))^(1/2)-1)-_C1 = 0