ODE
\[ a b x-a y'(x)^2+b y(x)+y'(x)^3=0 \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 1.22672 (sec), leaf count = 398
\[\text {Solve}\left [\left \{x=\frac {5 a \left (\frac {\sqrt [3]{2 a^3+\sqrt {\left (-2 a^3+27 a b x+27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{2 a^3+\sqrt {\left (-2 a^3+27 a b x+27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}+\frac {a}{3}\right )-\frac {3}{2} \left (\frac {\sqrt [3]{2 a^3+\sqrt {\left (-2 a^3+27 a b x+27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{2 a^3+\sqrt {\left (-2 a^3+27 a b x+27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}+\frac {a}{3}\right )^2-5 a^2 \log \left (\frac {\sqrt [3]{2 a^3+\sqrt {\left (-2 a^3+27 a b x+27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} a^2}{3 \sqrt [3]{2 a^3+\sqrt {\left (-2 a^3+27 a b x+27 b y(x)\right )^2-4 a^6}-27 a b x-27 b y(x)}}+\frac {4 a}{3}\right )}{b}+c_1\right \},y(x)\right ]\]
Maple ✓
cpu = 0.894 (sec), leaf count = 96
\[\left [y \left (x \right ) = -a x +\frac {a \left ({\mathrm e}^{\RootOf \left (-10 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+16 a \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_C1} b -13 a^{2}-2 b x \right )}-a \right )^{2}-\left ({\mathrm e}^{\RootOf \left (-10 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+16 a \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_C1} b -13 a^{2}-2 b x \right )}-a \right )^{3}}{b}\right ]\] Mathematica raw input
DSolve[a*b*x + b*y[x] - a*y'[x]^2 + y'[x]^3 == 0,y[x],x]
Mathematica raw output
Solve[{x == C[1] + (-5*a^2*Log[(4*a)/3 + (2^(1/3)*a^2)/(3*(2*a^3 - 27*a*b*x - 27*b*y[x] + Sqrt[-4*a^6 + (-2*a^3 + 27*a*b*x + 27*b*y[x])^2])^(1/3)) + (2*a^3 - 27*a*b*x - 27*b*y[x] + Sqrt[-4*a^6 + (-2*a^3 + 27*a*b*x + 27*b*y[x])^2])^(1/3)/(3*2^(1/3))] + 5*a*(a/3 + (2^(1/3)*a^2)/(3*(2*a^3 - 27*a*b*x - 27*b*y[x] + Sqrt[-4*a^6 + (-2*a^3 + 27*a*b*x + 27*b*y[x])^2])^(1/3)) + (2*a^3 - 27*a*b*x - 27*b*y[x] + Sqrt[-4*a^6 + (-2*a^3 + 27*a*b*x + 27*b*y[x])^2])^(1/3)/(3*2^(1/3))) - (3*(a/3 + (2^(1/3)*a^2)/(3*(2*a^3 - 27*a*b*x - 27*b*y[x] + Sqrt[-4*a^6 + (-2*a^3 + 27*a*b*x + 27*b*y[x])^2])^(1/3)) + (2*a^3 - 27*a*b*x - 27*b*y[x] + Sqrt[-4*a^6 + (-2*a^3 + 27*a*b*x + 27*b*y[x])^2])^(1/3)/(3*2^(1/3)))^2)/2)/b}, y[x]]
Maple raw input
dsolve(diff(y(x),x)^3-a*diff(y(x),x)^2+b*y(x)+a*b*x = 0, y(x))
Maple raw output
[y(x) = -a*x+(a*(exp(RootOf(-10*_Z*a^2-3*exp(2*_Z)+16*a*exp(_Z)+2*_C1*b-13*a^2-2*b*x))-a)^2-(exp(RootOf(-10*_Z*a^2-3*exp(2*_Z)+16*a*exp(_Z)+2*_C1*b-13*a^2-2*b*x))-a)^3)/b]