4.1.34 \(y'(x)=2 (y(x) \tan (2 x)+\sec (2 x)+1)\)

ODE
\[ y'(x)=2 (y(x) \tan (2 x)+\sec (2 x)+1) \] ODE Classification

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 0.0442997 (sec), leaf count = 20

\[\left \{\left \{y(x)\to \sec (2 x) \left (c_1+2 x+\sin (2 x)\right )\right \}\right \}\]

Maple
cpu = 0.485 (sec), leaf count = 100

\[ \left \{ y \relax (x ) ={\frac {1}{\sin \left (2\,x \right ) } \left (x\sin \left (4\,x \right ) \sqrt { \left (\cos \left (4\,x \right ) +1 \right ) ^{-1}}\sqrt {2\,\cos \left (4\,x \right ) +2}\sqrt {2+2\, \left (\tan \left (2\,x \right ) \right ) ^{2}}+\sqrt {1+ \left (\tan \left (2\,x \right ) \right ) ^{2}} \left ({\it \_C1}\,\sin \left (2\,x \right ) \sqrt {2\,\cos \left (4\,x \right ) +2}+{\frac {\cos \left (2\,x \right ) }{2}}-{\frac {\cos \left (6\,x \right ) }{2}} \right ) \right ) {\frac {1}{\sqrt {2\,\cos \left (4\,x \right ) +2}}}} \right \} \] Mathematica raw input

DSolve[y'[x] == 2*(1 + Sec[2*x] + Tan[2*x]*y[x]),y[x],x]

Mathematica raw output

{{y[x] -> Sec[2*x]*(2*x + C[1] + Sin[2*x])}}

Maple raw input

dsolve(diff(y(x),x) = 2+2*sec(2*x)+2*y(x)*tan(2*x), y(x),'implicit')

Maple raw output

y(x) = (x*sin(4*x)*(1/(cos(4*x)+1))^(1/2)*(2*cos(4*x)+2)^(1/2)*(2+2*tan(2*x)^2)^
(1/2)+(1+tan(2*x)^2)^(1/2)*(_C1*sin(2*x)*(2*cos(4*x)+2)^(1/2)+1/2*cos(2*x)-1/2*c
os(6*x)))/(2*cos(4*x)+2)^(1/2)/sin(2*x)