Internal problem ID [8326]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 746.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {i \left (i x +x^{4}+2 x^{2} y^{2}+y^{4}\right )}{y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.159 (sec). Leaf size: 232
dsolve(diff(y(x),x) = -I*(I*x+x^4+2*x^2*y(x)^2+y(x)^4)/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {2}\, \sqrt {\left (\AiryAi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1}+\AiryBi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right )\right ) \left (\left (1+i \sqrt {3}\right ) c_{1} \AiryAi \left (1, -\left (-8 i\right )^{\frac {1}{3}} x \right )+\left (1+i \sqrt {3}\right ) \AiryBi \left (1, -\left (-8 i\right )^{\frac {1}{3}} x \right )-2 x^{2} \left (\AiryAi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1}+\AiryBi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right )\right )\right )}}{2 \AiryAi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1}+2 \AiryBi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right )} \\ y \relax (x ) = \frac {\sqrt {2}\, \sqrt {\left (\AiryAi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1}+\AiryBi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right )\right ) \left (\left (1+i \sqrt {3}\right ) c_{1} \AiryAi \left (1, -\left (-8 i\right )^{\frac {1}{3}} x \right )+\left (1+i \sqrt {3}\right ) \AiryBi \left (1, -\left (-8 i\right )^{\frac {1}{3}} x \right )-2 x^{2} \left (\AiryAi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1}+\AiryBi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right )\right )\right )}}{2 \AiryAi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right ) c_{1}+2 \AiryBi \left (-\left (-8 i\right )^{\frac {1}{3}} x \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.21 (sec). Leaf size: 411
DSolve[y'[x] == ((-I)*(I*x + x^4 + 2*x^2*y[x]^2 + y[x]^4))/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\left (\text {Bi}\left (2 (-1)^{5/6} x\right )+c_1 \text {Ai}\left (2 (-1)^{5/6} x\right )\right ) \left (-2 x^2 \left (\text {Bi}\left (2 (-1)^{5/6} x\right )+c_1 \text {Ai}\left (2 (-1)^{5/6} x\right )\right )+\left (1+i \sqrt {3}\right ) \text {Bi}'\left (2 (-1)^{5/6} x\right )+\left (1+i \sqrt {3}\right ) c_1 \text {Ai}'\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \left (\text {Bi}\left (2 (-1)^{5/6} x\right )+c_1 \text {Ai}\left (2 (-1)^{5/6} x\right )\right )} \\ y(x)\to \frac {\sqrt {\left (\text {Bi}\left (2 (-1)^{5/6} x\right )+c_1 \text {Ai}\left (2 (-1)^{5/6} x\right )\right ) \left (-2 x^2 \left (\text {Bi}\left (2 (-1)^{5/6} x\right )+c_1 \text {Ai}\left (2 (-1)^{5/6} x\right )\right )+\left (1+i \sqrt {3}\right ) \text {Bi}'\left (2 (-1)^{5/6} x\right )+\left (1+i \sqrt {3}\right ) c_1 \text {Ai}'\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \left (\text {Bi}\left (2 (-1)^{5/6} x\right )+c_1 \text {Ai}\left (2 (-1)^{5/6} x\right )\right )} \\ y(x)\to -\frac {\sqrt {\text {Ai}\left (2 (-1)^{5/6} x\right ) \left (-2 x^2 \text {Ai}\left (2 (-1)^{5/6} x\right )+\left (1+i \sqrt {3}\right ) \text {Ai}'\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \text {Ai}\left (2 (-1)^{5/6} x\right )} \\ y(x)\to \frac {\sqrt {\text {Ai}\left (2 (-1)^{5/6} x\right ) \left (-2 x^2 \text {Ai}\left (2 (-1)^{5/6} x\right )+\left (1+i \sqrt {3}\right ) \text {Ai}'\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \text {Ai}\left (2 (-1)^{5/6} x\right )} \\ \end{align*}