##### 4.20.10 $$2 y(x) y'(x)^2+(5-4 x) y'(x)+2 y(x)=0$$

ODE
$2 y(x) y'(x)^2+(5-4 x) y'(x)+2 y(x)=0$ ODE Classiﬁcation

[[_homogeneous, class C], _rational, _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $$y$$

Mathematica
cpu = 0.348456 (sec), leaf count = 135

$\left \{\left \{y(x)\to -i \sqrt {2} e^{\frac {c_1}{2}} \sqrt {4 x-5+8 e^{c_1}}\right \},\left \{y(x)\to i \sqrt {2} e^{\frac {c_1}{2}} \sqrt {4 x-5+8 e^{c_1}}\right \},\left \{y(x)\to -\frac {1}{4} i e^{\frac {c_1}{2}} \sqrt {8 x-10+e^{c_1}}\right \},\left \{y(x)\to \frac {1}{4} i e^{\frac {c_1}{2}} \sqrt {8 x-10+e^{c_1}}\right \}\right \}$

Maple
cpu = 6.467 (sec), leaf count = 293

$\left [y \left (x \right ) = -x +\frac {5}{4}, y \left (x \right ) = x -\frac {5}{4}, \ln \left (x -\frac {5}{4}\right )+\ln \left (\frac {y \left (x \right )}{-5+4 x}\right )-\frac {\ln \left (\frac {4 y \left (x \right )}{-5+4 x}-1\right )}{2}-\frac {\ln \left (\frac {4 y \left (x \right )}{-5+4 x}+1\right )}{2}+\frac {\ln \left (\frac {16 y \left (x \right )^{2}}{\left (-5+4 x \right )^{2}}-1\right )}{2}-\sqrt {-\frac {16 y \left (x \right )^{2}}{\left (-5+4 x \right )^{2}}+1}+\arctanh \left (\frac {1}{\sqrt {-\frac {16 y \left (x \right )^{2}}{\left (-5+4 x \right )^{2}}+1}}\right )+\frac {\sqrt {4}\, \sqrt {\frac {-16 y \left (x \right )^{2}+16 x^{2}-40 x +25}{\left (-5+4 x \right )^{2}}}}{2}-\textit {\_C1} = 0, \ln \left (x -\frac {5}{4}\right )+\ln \left (\frac {y \left (x \right )}{-5+4 x}\right )-\frac {\ln \left (\frac {4 y \left (x \right )}{-5+4 x}-1\right )}{2}-\frac {\ln \left (\frac {4 y \left (x \right )}{-5+4 x}+1\right )}{2}+\sqrt {-\frac {16 y \left (x \right )^{2}}{\left (-5+4 x \right )^{2}}+1}-\arctanh \left (\frac {1}{\sqrt {-\frac {16 y \left (x \right )^{2}}{\left (-5+4 x \right )^{2}}+1}}\right )-\frac {\sqrt {4}\, \sqrt {\frac {-16 y \left (x \right )^{2}+16 x^{2}-40 x +25}{\left (-5+4 x \right )^{2}}}}{2}+\frac {\ln \left (\frac {16 y \left (x \right )^{2}}{\left (-5+4 x \right )^{2}}-1\right )}{2}-\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[2*y[x] + (5 - 4*x)*y'[x] + 2*y[x]*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-I)*Sqrt[2]*E^(C[1]/2)*Sqrt[-5 + 8*E^C[1] + 4*x]}, {y[x] -> I*Sqrt[2]
*E^(C[1]/2)*Sqrt[-5 + 8*E^C[1] + 4*x]}, {y[x] -> (-1/4*I)*E^(C[1]/2)*Sqrt[-10 +
E^C[1] + 8*x]}, {y[x] -> (I/4)*E^(C[1]/2)*Sqrt[-10 + E^C[1] + 8*x]}}

Maple raw input

dsolve(2*y(x)*diff(y(x),x)^2+(5-4*x)*diff(y(x),x)+2*y(x) = 0, y(x))

Maple raw output

[y(x) = -x+5/4, y(x) = x-5/4, ln(x-5/4)+ln(y(x)/(-5+4*x))-1/2*ln(4*y(x)/(-5+4*x)
-1)-1/2*ln(4*y(x)/(-5+4*x)+1)+1/2*ln(16*y(x)^2/(-5+4*x)^2-1)-(-16*y(x)^2/(-5+4*x
)^2+1)^(1/2)+arctanh(1/(-16*y(x)^2/(-5+4*x)^2+1)^(1/2))+1/2*4^(1/2)*((-16*y(x)^2
+16*x^2-40*x+25)/(-5+4*x)^2)^(1/2)-_C1 = 0, ln(x-5/4)+ln(y(x)/(-5+4*x))-1/2*ln(4
*y(x)/(-5+4*x)-1)-1/2*ln(4*y(x)/(-5+4*x)+1)+(-16*y(x)^2/(-5+4*x)^2+1)^(1/2)-arct
anh(1/(-16*y(x)^2/(-5+4*x)^2+1)^(1/2))-1/2*4^(1/2)*((-16*y(x)^2+16*x^2-40*x+25)/
(-5+4*x)^2)^(1/2)+1/2*ln(16*y(x)^2/(-5+4*x)^2-1)-_C1 = 0]