4.18.21 \(a x+x y'(x)^2-2 y(x) y'(x)=0\)

ODE
\[ a x+x y'(x)^2-2 y(x) y'(x)=0 \] ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Homogeneous ODE, \(x^n f\left (\frac {y}{x} , y' \right )=0\), Solve for \(y\)

Mathematica
cpu = 0.072401 (sec), leaf count = 151

\[\left \{\left \{y(x)\to -\frac {\sqrt {a} x \tan \left (c_1-i \log (x)\right )}{\sqrt {\sec ^2\left (c_1-i \log (x)\right )}}\right \},\left \{y(x)\to \frac {\sqrt {a} x \tan \left (c_1-i \log (x)\right )}{\sqrt {\sec ^2\left (c_1-i \log (x)\right )}}\right \},\left \{y(x)\to -\frac {\sqrt {a} x \tan \left (c_1+i \log (x)\right )}{\sqrt {\sec ^2\left (c_1+i \log (x)\right )}}\right \},\left \{y(x)\to \frac {\sqrt {a} x \tan \left (c_1+i \log (x)\right )}{\sqrt {\sec ^2\left (c_1+i \log (x)\right )}}\right \}\right \}\]

Maple
cpu = 0.023 (sec), leaf count = 32

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-a{x}^{2}=0,[x \left ({\it \_T} \right ) ={\it \_T}\,{\it \_C1},y \left ({\it \_T} \right ) ={\frac { \left ({{\it \_T}}^{2}+a \right ) {\it \_C1}}{2}}] \right \} \] Mathematica raw input

DSolve[a*x - 2*y[x]*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((Sqrt[a]*x*Tan[C[1] - I*Log[x]])/Sqrt[Sec[C[1] - I*Log[x]]^2])}, {y[
x] -> (Sqrt[a]*x*Tan[C[1] - I*Log[x]])/Sqrt[Sec[C[1] - I*Log[x]]^2]}, {y[x] -> -
((Sqrt[a]*x*Tan[C[1] + I*Log[x]])/Sqrt[Sec[C[1] + I*Log[x]]^2])}, {y[x] -> (Sqrt
[a]*x*Tan[C[1] + I*Log[x]])/Sqrt[Sec[C[1] + I*Log[x]]^2]}}

Maple raw input

dsolve(x*diff(y(x),x)^2-2*y(x)*diff(y(x),x)+a*x = 0, y(x),'implicit')

Maple raw output

y(x)^2-a*x^2 = 0, [x(_T) = _T*_C1, y(_T) = 1/2*(_T^2+a)*_C1]