4.16.49 \(y'(x)^2+2 (1-x) y'(x)-2 (x-y(x))=0\)

ODE
\[ y'(x)^2+2 (1-x) y'(x)-2 (x-y(x))=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _dAlembert]

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

Mathematica
cpu = 1.3226 (sec), leaf count = 287

\[\left \{\text {Solve}\left [2 c_1+2 \sqrt {x^2-2 y(x)+1}+\log \left (\sqrt {2} \sqrt {y(x)}-x\right )+\log \left (\sqrt {2} \sqrt {y(x)}+x\right )+\log (1-2 y(x))=\log \left (x^2-2 y(x)\right )+\log \left (\sqrt {x^2-2 y(x)+1}-\sqrt {2} x \sqrt {y(x)}-2 y(x)+1\right )+\log \left (\sqrt {x^2-2 y(x)+1}+\sqrt {2} x \sqrt {y(x)}-2 y(x)+1\right )+2 x,y(x)\right ],\text {Solve}\left [2 \sqrt {x^2-2 y(x)+1}+\log \left (x^2-2 y(x)\right )+\log \left (\sqrt {2} \sqrt {y(x)}-x\right )+\log \left (\sqrt {2} \sqrt {y(x)}+x\right )+\log (2 y(x)-1)+2 x=2 c_1+\log \left (\sqrt {x^2-2 y(x)+1}-\sqrt {2} x \sqrt {y(x)}-2 y(x)+1\right )+\log \left (\sqrt {x^2-2 y(x)+1}+\sqrt {2} x \sqrt {y(x)}-2 y(x)+1\right ),y(x)\right ]\right \}\]

Maple
cpu = 0.016 (sec), leaf count = 42

\[ \left \{ [x \left ({\it \_T} \right ) ={{\rm e}^{-{\it \_T}}} \left ({{\rm e}^{{\it \_T}}}{\it \_T}+{\it \_C1} \right ) ,y \left ({\it \_T} \right ) = \left (1+{\it \_T} \right ) {{\rm e}^{-{\it \_T}}} \left ({{\rm e}^{{\it \_T}}}{\it \_T}+{\it \_C1} \right ) -{\frac {{{\it \_T}}^{2}}{2}}-{\it \_T}] \right \} \] Mathematica raw input

DSolve[-2*(x - y[x]) + 2*(1 - x)*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[2*C[1] + Log[-x + Sqrt[2]*Sqrt[y[x]]] + Log[x + Sqrt[2]*Sqrt[y[x]]] + Log
[1 - 2*y[x]] + 2*Sqrt[1 + x^2 - 2*y[x]] == 2*x + Log[x^2 - 2*y[x]] + Log[1 + Sqr
t[1 + x^2 - 2*y[x]] - Sqrt[2]*x*Sqrt[y[x]] - 2*y[x]] + Log[1 + Sqrt[1 + x^2 - 2*
y[x]] + Sqrt[2]*x*Sqrt[y[x]] - 2*y[x]], y[x]], Solve[2*x + Log[-x + Sqrt[2]*Sqrt
[y[x]]] + Log[x + Sqrt[2]*Sqrt[y[x]]] + Log[x^2 - 2*y[x]] + Log[-1 + 2*y[x]] + 2
*Sqrt[1 + x^2 - 2*y[x]] == 2*C[1] + Log[1 + Sqrt[1 + x^2 - 2*y[x]] - Sqrt[2]*x*S
qrt[y[x]] - 2*y[x]] + Log[1 + Sqrt[1 + x^2 - 2*y[x]] + Sqrt[2]*x*Sqrt[y[x]] - 2*
y[x]], y[x]]}

Maple raw input

dsolve(diff(y(x),x)^2+2*(1-x)*diff(y(x),x)-2*x+2*y(x) = 0, y(x),'implicit')

Maple raw output

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