Internal problem ID [9104]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 769.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {y^{\prime }+\frac {i \left (16 i x^{2}+16 y^{4}+8 y^{2} x^{4}+x^{8}\right ) x}{32 y}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 251
dsolve(diff(y(x),x) = -1/32*I*(16*I*x^2+16*y(x)^4+8*x^4*y(x)^2+x^8)*x/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {\sqrt {4}\, \sqrt {\left (-2 c_{1} \left (\frac {x^{6}}{8}+i\right ) \operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\left (-\frac {x^{6}}{4}-2 i\right ) \operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\left (1+i\right ) \left (\operatorname {BesselJ}\left (\frac {4}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {4}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x^{3}\right ) \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right )}}{2 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x} y \left (x \right ) = \frac {\sqrt {4}\, \sqrt {\left (-2 c_{1} \left (\frac {x^{6}}{8}+i\right ) \operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\left (-\frac {x^{6}}{4}-2 i\right ) \operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )+\left (1+i\right ) \left (\operatorname {BesselJ}\left (\frac {4}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {4}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x^{3}\right ) \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right )}}{2 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {1}{3}-\frac {i}{3}\right ) x^{3}\right )\right ) x} \end{align*}
✓ Solution by Mathematica
Time used: 39.169 (sec). Leaf size: 836
DSolve[y'[x] == ((-1/32*I)*x*((16*I)*x^2 + x^8 + 8*x^4*y[x]^2 + 16*y[x]^4))/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right ) \left ((1+i) x^3 \left (\operatorname {BesselY}\left (\frac {4}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {4}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )-\frac {1}{4} \left (x^6+8 i\right ) \operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )-\frac {1}{4} c_1 \left (x^6+8 i\right ) \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )}}{x \left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )} y(x)\to \frac {\sqrt {\left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right ) \left ((1+i) x^3 \left (\operatorname {BesselY}\left (\frac {4}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {4}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )-\frac {1}{4} \left (x^6+8 i\right ) \operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )-\frac {1}{4} c_1 \left (x^6+8 i\right ) \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )}}{x \left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {1}{3}-\frac {i}{3}\right ) x^3\right )\right )} y(x)\to -\frac {(-1)^{5/6} x \sqrt {-\frac {\sqrt [6]{-1} \left ((1-i) x^3\right )^{2/3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right ) \left (-4 i 2^{2/3} \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )+4 i 2^{2/3} \operatorname {AiryBiPrime}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )+\sqrt [6]{-1} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\sqrt [6]{-1} x^4 \operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right )}{x^2}}}{2 \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right )} y(x)\to \frac {(-1)^{5/6} x \sqrt {-\frac {\sqrt [6]{-1} \left ((1-i) x^3\right )^{2/3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right ) \left (-4 i 2^{2/3} \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )+4 i 2^{2/3} \operatorname {AiryBiPrime}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )+\sqrt [6]{-1} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\sqrt [6]{-1} x^4 \operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right )}{x^2}}}{2 \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )-\operatorname {AiryBi}\left (\frac {(-1)^{5/6} x^2}{\sqrt [3]{2}}\right )\right )} \end{align*}