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 = 0.362566 (sec), leaf count = 155
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {(3 a+2 b) \log \left (-3 \sqrt {a^2-4 \text {$\#$1} b}+3 a+2 b\right )+3 a \log \left (\sqrt {a^2-4 \text {$\#$1} b}+a\right )}{6 (3 a+b)}\& \right ]\left [-\frac {x}{2}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 a \log \left (a-\sqrt {a^2-4 \text {$\#$1} b}\right )+(3 a+2 b) \log \left (3 \sqrt {a^2-4 \text {$\#$1} b}+3 a+2 b\right )}{6 (3 a+b)}\& \right ]\left [\frac {x}{2}+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.109 (sec), leaf count = 71
\[\left [x -\left (\int _{}^{y \left (x \right )}\frac {1}{-\frac {a}{2}-3 \textit {\_a} -\frac {\sqrt {-4 b \textit {\_a} +a^{2}}}{2}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}\frac {1}{-\frac {a}{2}-3 \textit {\_a} +\frac {\sqrt {-4 b \textit {\_a} +a^{2}}}{2}}d \textit {\_a} \right )-\textit {\_C1} = 0\right ]\] 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[((3*a + 2*b)*Log[3*a + 2*b - 3*Sqrt[a^2 - 4*b*#1]] + 3*a*Log[a + Sqrt[a^2 - 4*b*#1]])/(6*(3*a + b)) & ][-1/2*x + C[1]]}, {y[x] -> InverseFunction[-1/6*(3*a*Log[a - Sqrt[a^2 - 4*b*#1]] + (3*a + 2*b)*Log[3*a + 2*b + 3*Sqrt[a^2 - 4*b*#1]])/(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))
Maple raw output
[x-Intat(1/(-1/2*a-3*_a-1/2*(-4*_a*b+a^2)^(1/2)),_a = y(x))-_C1 = 0, x-Intat(1/(-1/2*a-3*_a+1/2*(-4*_a*b+a^2)^(1/2)),_a = y(x))-_C1 = 0]