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 = 0.225076 (sec), leaf count = 135

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

Maple
cpu = 0.587 (sec), leaf count = 152

\[ \left \{ \left (y \relax (x ) \right ) ^{2}- \left (x-{\frac {5}{4}} \right ) ^{2}=0,\ln \left (x-{\frac {5}{4}} \right ) +{\frac {1}{2}\ln \left (16\,{\frac { \left (y \relax (x ) \right ) ^{2}}{ \left (-5+4\,x \right ) ^{2}}}-1 \right ) }-{\frac {\sqrt {4}}{2}\sqrt {{\frac {1}{ \left (-5+4\,x \right ) ^{2}} \left (-16\, \left (y \relax (x ) \right ) ^{2}+16\, \left (x-5/4 \right ) ^{2} \right ) }}}+\sqrt {-16\,{\frac { \left (y \relax (x ) \right ) ^{2}}{ \left (-5+4\,x \right ) ^{2}}}+1}-{\it Artanh} \left ({\frac {1}{\sqrt {-16\,{\frac { \left (y \relax (x ) \right ) ^{2}}{ \left (-5+4\,x \right ) ^{2}}}+1}}} \right ) -{\frac {1}{2}\ln \left (4\,{\frac {y \relax (x ) }{-5+4\,x}}+1 \right ) }-{\frac {1}{2}\ln \left (4\,{\frac {y \relax (x ) }{-5+4\,x}}-1 \right ) }+\ln \left ({\frac {y \relax (x ) }{-5+4\,x}} \right ) -{\it \_C1}=0 \right \} \] 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] -> (-I/4)*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),'implicit')

Maple raw output

y(x)^2-(x-5/4)^2 = 0, ln(x-5/4)+1/2*ln(16*y(x)^2/(-5+4*x)^2-1)-1/2*4^(1/2)*((-16
*y(x)^2+16*(x-5/4)^2)/(-5+4*x)^2)^(1/2)+(-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*ln(4*y(x)/(-5+4*x)+1)-1/2*ln(4*y(x)/(-5+4
*x)-1)+ln(y(x)/(-5+4*x))-_C1 = 0