4.5.40 \(2 x y'(x)+1=y(x)^2+4 i x y(x)\)

ODE
\[ 2 x y'(x)+1=y(x)^2+4 i x y(x) \] ODE Classification

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica ✓
cpu = 0.104555 (sec), leaf count = 124

\[\left \{\left \{y(x)\to \frac {(1-i) \sqrt {x} \left ((2-2 i) \sqrt {x} G_{1,2}^{2,0}\left (-2 i x\left |\begin {array}{c} -1 \\ -\frac {3}{2},-\frac {1}{2} \\\end {array}\right .\right )+c_1 e^{i x} ((x-i) J_0(x)-J_1(x)+x J_2(x))\right )}{G_{1,2}^{2,0}\left (-2 i x\left |\begin {array}{c} 1 \\ -\frac {1}{2},\frac {1}{2} \\\end {array}\right .\right )+(1+i) c_1 e^{i x} \sqrt {x} (J_0(x)-i J_1(x))}\right \}\right \}\]

Maple ✓
cpu = 0.253 (sec), leaf count = 62

\[ \left \{ y \left ( x \right ) ={\frac {-{{\sl K}_{0}\left (ix\right )}{\it \_C1}+i{{\sl J}_{1}\left (x\right )}+{{\sl K}_{1}\left (ix\right )}{\it \_C1}+{{\sl J}_{0}\left (x\right )}}{{{\sl K}_{0}\left (ix\right )}{\it \_C1}+i{{\sl J}_{1}\left (x\right )}+{{\sl K}_{1}\left (ix\right )}{\it \_C1}-{{\sl J}_{0}\left (x\right )}}} \right \} \] Mathematica raw input

DSolve[1 + 2*x*y'[x] == (4*I)*x*y[x] + y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> ((1 - I)*Sqrt[x]*(E^(I*x)*((-I + x)*BesselJ[0, x] - BesselJ[1, x] + x*
BesselJ[2, x])*C[1] + (2 - 2*I)*Sqrt[x]*MeijerG[{{}, {-1}}, {{-3/2, -1/2}, {}}, 
(-2*I)*x]))/((1 + I)*E^(I*x)*Sqrt[x]*(BesselJ[0, x] - I*BesselJ[1, x])*C[1] + Me
ijerG[{{}, {1}}, {{-1/2, 1/2}, {}}, (-2*I)*x])}}

Maple raw input

dsolve(2*x*diff(y(x),x)+1 = 4*I*x*y(x)+y(x)^2, y(x),'implicit')

Maple raw output

y(x) = (-BesselK(0,I*x)*_C1+I*BesselJ(1,x)+BesselK(1,I*x)*_C1+BesselJ(0,x))/(Bes
selK(0,I*x)*_C1+I*BesselJ(1,x)+BesselK(1,I*x)*_C1-BesselJ(0,x))