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 = 0.17051 (sec), leaf count = 59
\[\left \{\left \{y(x)\to \frac {1}{4} x^2 \left (\sqrt {a^2+4 b c}-a\right )+c_1\right \},\left \{y(x)\to -\frac {1}{4} x^2 \left (\sqrt {a^2+4 b c}+a\right )+c_1\right \}\right \}\]
Maple ✓
cpu = 0.035 (sec), leaf count = 47
\[\left [y \left (x \right ) = \frac {\left (-a +\sqrt {a^{2}+4 c b}\right ) x^{2}}{4}+\textit {\_C1}, y \left (x \right ) = -\frac {\left (a +\sqrt {a^{2}+4 c b}\right ) x^{2}}{4}+\textit {\_C1}\right ]\] 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] -> -1/4*((a + Sqrt[a^2 + 4*b*c])*x^2) + C[1]}}
Maple raw input
dsolve(diff(y(x),x)^2+a*x*diff(y(x),x) = b*c*x^2, y(x))
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]