Internal problem ID [2985]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 8
Problem number: 237.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {2 y^{\prime } x +1-4 i x y-y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 64
dsolve(2*x*diff(y(x),x)+1 = 4*I*x*y(x)+y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {i \BesselJ \left (1, x\right )-\BesselK \left (1, i x \right ) c_{1}+\BesselK \left (0, i x \right ) c_{1}+\BesselJ \left (0, x\right )}{i \BesselJ \left (1, x\right )-\BesselK \left (1, i x \right ) c_{1}-\BesselK \left (0, i x \right ) c_{1}-\BesselJ \left (0, x\right )} \]
✓ Solution by Mathematica
Time used: 0.386 (sec). Leaf size: 202
DSolve[2 x y'[x]+1==4 I x y[x]+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(1-i) c_1 e^{i x} \sqrt {x} ((x-i) J_0(x)-J_1(x)+x J_2(x))-4 i x G_{1,2}^{2,0}\left (-2 i x\left | {c} -1 \\ -\frac {3}{2},-\frac {1}{2} \\ \\ \right .\right )}{G_{1,2}^{2,0}\left (-2 i x\left | {c} 1 \\ -\frac {1}{2},\frac {1}{2} \\ \\ \right .\right )+(1+i) c_1 e^{i x} \sqrt {x} (J_0(x)-i J_1(x))} \\ y(x)\to -\frac {i ((x-i) J_0(x)-J_1(x)+x J_2(x))}{J_0(x)-i J_1(x)} \\ y(x)\to -\frac {i ((x-i) J_0(x)-J_1(x)+x J_2(x))}{J_0(x)-i J_1(x)} \\ \end{align*}