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

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

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

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

Mathematica ✓
cpu = 1.12971 (sec), leaf count = 163

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

Maple ✓
cpu = 0.021 (sec), leaf count = 44

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

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

Mathematica raw output

{Solve[((-1 + Sqrt[-1 + (2*y[x])/x])*(-1 + Log[-1 + Sqrt[-1 + (2*y[x])/x]]*(-1 +
 Sqrt[-1 + (2*y[x])/x])))/(-(y[x]/x) + Sqrt[-1 + (2*y[x])/x]) == C[1] + Log[x], 
y[x]], Solve[Log[x] + (x*(1 + Sqrt[-1 + (2*y[x])/x])*(1 + Log[1 + Sqrt[-1 + (2*y
[x])/x]]*(1 + Sqrt[-1 + (2*y[x])/x])))/(y[x] + x*Sqrt[-1 + (2*y[x])/x]) == C[1],
 y[x]]}

Maple raw input

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

Maple raw output

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