ODE
\[ a x y'(x)+y'(x)^2=b c x^2 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.0150139 (sec), leaf count = 59
\[\left \{\left \{y(x)\to \frac {1}{4} x^2 \left (\sqrt {a^2+4 b c}-a\right )+c_1\right \},\left \{y(x)\to c_1-\frac {1}{4} x^2 \left (\sqrt {a^2+4 b c}+a\right )\right \}\right \}\]
Maple ✓
cpu = 0.026 (sec), leaf count = 47
\[ \left \{ y \left ( x \right ) ={\frac {{x}^{2}}{4} \left ( -a+\sqrt {{a}^{2}+4\,cb} \right ) }+{\it \_C1},y \left ( x \right ) =-{\frac {{x}^{2}}{4} \left ( a+\sqrt {{a}^{2}+4\,cb} \right ) }+{\it \_C1} \right \} \] Mathematica raw input
DSolve[a*x*y'[x] + y'[x]^2 == b*c*x^2,y[x],x]
Mathematica raw output
{{y[x] -> ((-a + Sqrt[a^2 + 4*b*c])*x^2)/4 + C[1]}, {y[x] -> -((a + Sqrt[a^2 + 4*b*c])*x^2)/4 + C[1]}}
Maple raw input
dsolve(diff(y(x),x)^2+a*x*diff(y(x),x) = b*c*x^2, y(x),'implicit')
Maple raw output
y(x) = 1/4*(-a+(a^2+4*b*c)^(1/2))*x^2+_C1, y(x) = -1/4*(a+(a^2+4*b*c)^(1/2))*x^2+_C1