Internal problem ID [9094]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 759.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {y^{\prime }+\frac {i \left (54 i x^{2}+81 y^{4}+18 x^{4} y^{2}+x^{8}\right ) x}{243 y}=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 283
dsolve(diff(y(x),x) = -1/243*I*(54*I*x^2+81*y(x)^4+18*x^4*y(x)^2+x^8)*x/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-3 x^{3} \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \left (\frac {\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} x^{3}}{3}+\frac {\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) x^{3}}{3}+\left (1+i\right ) \left (\operatorname {BesselJ}\left (-\frac {2}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (-\frac {2}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \sqrt {6}\right )}}{3 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) x} \\ y \left (x \right ) &= \frac {\sqrt {-3 x^{3} \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \left (\frac {\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} x^{3}}{3}+\frac {\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) x^{3}}{3}+\left (1+i\right ) \left (\operatorname {BesselJ}\left (-\frac {2}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (-\frac {2}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) \sqrt {6}\right )}}{3 \left (\operatorname {BesselJ}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{3}, \left (\frac {2}{27}-\frac {2 i}{27}\right ) \sqrt {6}\, x^{3}\right )\right ) x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 37.777 (sec). Leaf size: 1293
DSolve[y'[x] == ((-1/243*I)*x*((54*I)*x^2 + x^8 + 18*x^4*y[x]^2 + 81*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 {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right ) \left ((1+i) \sqrt {6} x^3 \left (\operatorname {BesselY}\left (\frac {4}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {4}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )-\frac {1}{3} \left (x^6+27 i\right ) \operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )-\frac {1}{3} c_1 \left (x^6+27 i\right ) \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )}}{\sqrt {3} x \left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )} \\ y(x)\to \frac {\sqrt {\left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right ) \left ((1+i) \sqrt {6} x^3 \left (\operatorname {BesselY}\left (\frac {4}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {4}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )-\frac {1}{3} \left (x^6+27 i\right ) \operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )-\frac {1}{3} c_1 \left (x^6+27 i\right ) \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )\right )}}{\sqrt {3} x \left (\operatorname {BesselY}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} x^3\right )+c_1 \operatorname {BesselJ}\left (\frac {1}{3},\left (\frac {2}{9}-\frac {2 i}{9}\right ) \sqrt {\frac {2}{3}} 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}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right ) \left (-18 i \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+18 i \operatorname {AiryBiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+\sqrt [6]{-1} \sqrt [3]{2} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\sqrt [6]{-1} \sqrt [3]{2} x^4 \operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )}{x^2}}}{3 \sqrt [6]{2} \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^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}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right ) \left (-18 i \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+18 i \operatorname {AiryBiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+\sqrt [6]{-1} \sqrt [3]{2} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\sqrt [6]{-1} \sqrt [3]{2} x^4 \operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )}{x^2}}}{3 \sqrt [6]{2} \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^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}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right ) \left (-18 i \sqrt {3} \operatorname {AiryAiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+18 i \operatorname {AiryBiPrime}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )+\sqrt [6]{-1} \sqrt [3]{2} \sqrt {3} x^4 \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\sqrt [6]{-1} \sqrt [3]{2} x^4 \operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )}{x^2}}}{3 \sqrt [6]{2} \sqrt [3]{(1-i) x^3} \left (\sqrt {3} \operatorname {AiryAi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )-\operatorname {AiryBi}\left (\frac {1}{3} (-1)^{5/6} 2^{2/3} x^2\right )\right )} \\ \end{align*}