29.8.32 problem 237

Internal problem ID [4837]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 8
Problem number : 237
Date solved : Monday, January 27, 2025 at 09:42:05 AM
CAS classification : [_rational, _Riccati]

\begin{align*} 2 x y^{\prime }+1&=4 i x y+y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 58

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: 0.488 (sec). Leaf size: 161

DSolve[2 x D[y[x],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)} \\ \end{align*}