Internal problem ID [3493]
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]
\[ \boxed {2 x y^{\prime }-4 i x y-y^{2}=-1} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 62
dsolve(2*x*diff(y(x),x)+1 = 4*I*x*y(x)+y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {i \operatorname {BesselJ}\left (1, x\right )+\operatorname {BesselK}\left (1, i x \right ) c_{1} -\operatorname {BesselK}\left (0, i x \right ) c_{1} +\operatorname {BesselJ}\left (0, x\right )}{i \operatorname {BesselJ}\left (1, x\right )+\operatorname {BesselK}\left (1, i x \right ) c_{1} +\operatorname {BesselK}\left (0, i x \right ) c_{1} -\operatorname {BesselJ}\left (0, x\right )} \]
✓ Solution by Mathematica
Time used: 0.538 (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) \operatorname {BesselJ}(0,x)-\operatorname {BesselJ}(1,x)+x \operatorname {BesselJ}(2,x))-4 i 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 )}{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} (\operatorname {BesselJ}(0,x)-i \operatorname {BesselJ}(1,x))} y(x)\to -\frac {i ((x-i) \operatorname {BesselJ}(0,x)-\operatorname {BesselJ}(1,x)+x \operatorname {BesselJ}(2,x))}{\operatorname {BesselJ}(0,x)-i \operatorname {BesselJ}(1,x)} y(x)\to -\frac {i ((x-i) \operatorname {BesselJ}(0,x)-\operatorname {BesselJ}(1,x)+x \operatorname {BesselJ}(2,x))}{\operatorname {BesselJ}(0,x)-i \operatorname {BesselJ}(1,x)} \end{align*}