Internal problem ID [9078]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 743.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {y^{\prime }+\frac {i \left (8 i x +16 y^{4}+8 y^{2} x^{2}+x^{4}\right )}{32 y}=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 296
dsolve(diff(y(x),x) = -1/32*I*(8*I*x+16*y(x)^4+8*x^2*y(x)^2+x^4)/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {\sqrt {2}\, \sqrt {\left (\operatorname {AiryAi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right ) c_{1} +\operatorname {AiryBi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right )\right ) \left (\left (1+i \sqrt {3}\right ) c_{1} \operatorname {AiryAi}\left (1, \frac {\left (i-\sqrt {3}\right ) x}{2}\right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBi}\left (1, \frac {\left (i-\sqrt {3}\right ) x}{2}\right )-\frac {x^{2} \left (\operatorname {AiryAi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right ) c_{1} +\operatorname {AiryBi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right )\right )}{2}\right )}}{2 \operatorname {AiryAi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right ) c_{1} +2 \operatorname {AiryBi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right )} y \left (x \right ) = \frac {\sqrt {2}\, \sqrt {\left (\operatorname {AiryAi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right ) c_{1} +\operatorname {AiryBi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right )\right ) \left (\left (1+i \sqrt {3}\right ) c_{1} \operatorname {AiryAi}\left (1, \frac {\left (i-\sqrt {3}\right ) x}{2}\right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBi}\left (1, \frac {\left (i-\sqrt {3}\right ) x}{2}\right )-\frac {x^{2} \left (\operatorname {AiryAi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right ) c_{1} +\operatorname {AiryBi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right )\right )}{2}\right )}}{2 \operatorname {AiryAi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right ) c_{1} +2 \operatorname {AiryBi}\left (\frac {\left (i-\sqrt {3}\right ) x}{2}\right )} \end{align*}
✓ Solution by Mathematica
Time used: 5.963 (sec). Leaf size: 553
DSolve[y'[x] == ((-1/32*I)*((8*I)*x + x^4 + 8*x^2*y[x]^2 + 16*y[x]^4))/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-\left (\left (\operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right ) \left (x^2 \operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \left (x^2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )-2 i \left (\sqrt {3}-i\right ) \operatorname {AiryAiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )+\left (-2-2 i \sqrt {3}\right ) \operatorname {AiryBiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )\right )}}{2 \left (\operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )} y(x)\to \frac {\sqrt {-\left (\left (\operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right ) \left (x^2 \operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \left (x^2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )-2 i \left (\sqrt {3}-i\right ) \operatorname {AiryAiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )+\left (-2-2 i \sqrt {3}\right ) \operatorname {AiryBiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )\right )}}{2 \left (\operatorname {AiryBi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+c_1 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )} y(x)\to -\frac {\sqrt {-\operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right ) \left (x^2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )-2 i \left (\sqrt {3}-i\right ) \operatorname {AiryAiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )}}{2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )} y(x)\to \frac {\sqrt {-\operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right ) \left (x^2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )-2 i \left (\sqrt {3}-i\right ) \operatorname {AiryAiPrime}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )\right )}}{2 \operatorname {AiryAi}\left (-\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )} \end{align*}