4.18.39 \(4 x y'(x)^2+2 x y'(x)-y(x)=0\)

ODE
\[ 4 x y'(x)^2+2 x y'(x)-y(x)=0 \] ODE Classification

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

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 0.0808966 (sec), leaf count = 58

\[\left \{\left \{y(x)\to \frac {1}{4} e^{2 c_1} \left (e^{2 c_1}-2 \sqrt {x}\right )\right \},\left \{y(x)\to \frac {1}{4} e^{-4 c_1} \left (1-2 e^{2 c_1} \sqrt {x}\right )\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 29

\[ \left \{ y \relax (x ) =-{\frac {x}{4}},[x \left ({\it \_T} \right ) ={\frac {{\it \_C1}}{{{\it \_T}}^{2}}},y \left ({\it \_T} \right ) ={\frac { \left (4\,{\it \_T}+2 \right ) {\it \_C1}}{{\it \_T}}}] \right \} \] Mathematica raw input

DSolve[-y[x] + 2*x*y'[x] + 4*x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (E^(2*C[1])*(E^(2*C[1]) - 2*Sqrt[x]))/4}, {y[x] -> (1 - 2*E^(2*C[1])*S
qrt[x])/(4*E^(4*C[1]))}}

Maple raw input

dsolve(4*x*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = -1/4*x, [x(_T) = 1/_T^2*_C1, y(_T) = (4*_T+2)*_C1/_T]