ODE
\[ y(x) (3 a+b+9 y(x))+(a+6 y(x)) y'(x)+y'(x)^2=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.346644 (sec), leaf count = 226
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {6 a \tanh ^{-1}\left (\frac {\sqrt {a^2-4 \text {$\#$1} b}}{a}\right )-2 (3 a+2 b) \tanh ^{-1}\left (\frac {3 \sqrt {a^2-4 \text {$\#$1} b}}{3 a+2 b}\right )+3 a \log (9 \text {$\#$1}+3 a+b)+2 b \log (9 \text {$\#$1}+3 a+b)+3 a \log (\text {$\#$1})}{12 (3 a+b)}\& \right ]\left [c_1-\frac {x}{2}\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {-6 a \tanh ^{-1}\left (\frac {\sqrt {a^2-4 \text {$\#$1} b}}{a}\right )+(6 a+4 b) \tanh ^{-1}\left (\frac {3 \sqrt {a^2-4 \text {$\#$1} b}}{3 a+2 b}\right )+3 a \log (9 \text {$\#$1}+3 a+b)+2 b \log (9 \text {$\#$1}+3 a+b)+3 a \log (\text {$\#$1})}{12 (3 a+b)}\& \right ]\left [c_1+\frac {x}{2}\right ]\right \}\right \}\]
Maple ✓
cpu = 0.067 (sec), leaf count = 69
\[ \left \{ x-\int ^{y \left ( x \right ) }\!-2\, \left ( a+6\,{\it \_a}-\sqrt {-4\,{\it \_a}\,b+{a}^{2}} \right ) ^{-1}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!-2\, \left ( a+6\,{\it \_a}+\sqrt {-4\,{\it \_a}\,b+{a}^{2}} \right ) ^{-1}{d{\it \_a}}-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y[x]*(3*a + b + 9*y[x]) + (a + 6*y[x])*y'[x] + y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(6*a*ArcTanh[Sqrt[a^2 - 4*b*#1]/a] - 2*(3*a + 2*b)*ArcTanh[(3*Sqrt[a^2 - 4*b*#1])/(3*a + 2*b)] + 3*a*Log[#1] + 3*a*Log[3*a + b + 9*#1] + 2*b*Log[3*a + b + 9*#1])/(12*(3*a + b)) & ][-x/2 + C[1]]}, {y[x] -> InverseFunction[-(-6*a*ArcTanh[Sqrt[a^2 - 4*b*#1]/a] + (6*a + 4*b)*ArcTanh[(3*Sqrt[a^2 - 4*b*#1])/(3*a + 2*b)] + 3*a*Log[#1] + 3*a*Log[3*a + b + 9*#1] + 2*b*Log[3*a + b + 9*#1])/(12*(3*a + b)) & ][x/2 + C[1]]}}
Maple raw input
dsolve(diff(y(x),x)^2+(a+6*y(x))*diff(y(x),x)+y(x)*(3*a+b+9*y(x)) = 0, y(x),'implicit')
Maple raw output
x-Intat(-2/(a+6*_a+(-4*_a*b+a^2)^(1/2)),_a = y(x))-_C1 = 0, x-Intat(-2/(a+6*_a-(-4*_a*b+a^2)^(1/2)),_a = y(x))-_C1 = 0