60.2.170 problem 746
Internal
problem
ID
[10757]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
746
Date
solved
:
Tuesday, January 28, 2025 at 05:11:43 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=-\frac {i \left (i x +x^{4}+2 x^{2} y^{2}+y^{4}\right )}{y} \end{align*}
✓ Solution by Maple
Time used: 0.083 (sec). Leaf size: 212
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 &= -\frac {\sqrt {2}\, \sqrt {\left (\operatorname {AiryAi}\left (-\left (-8 i\right )^{{1}/{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (-\left (-8 i\right )^{{1}/{3}} x \right )\right ) \left (\left (1+i \sqrt {3}\right ) c_{1} \operatorname {AiryAi}\left (1, -\left (-8 i\right )^{{1}/{3}} x \right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBi}\left (1, -\left (-8 i\right )^{{1}/{3}} x \right )-2 x^{2} \left (\operatorname {AiryAi}\left (-\left (-8 i\right )^{{1}/{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (-\left (-8 i\right )^{{1}/{3}} x \right )\right )\right )}}{2 \operatorname {AiryAi}\left (-\left (-8 i\right )^{{1}/{3}} x \right ) c_{1} +2 \operatorname {AiryBi}\left (-\left (-8 i\right )^{{1}/{3}} x \right )} \\
y &= \frac {\sqrt {2}\, \sqrt {\left (\operatorname {AiryAi}\left (-\left (-8 i\right )^{{1}/{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (-\left (-8 i\right )^{{1}/{3}} x \right )\right ) \left (\left (1+i \sqrt {3}\right ) c_{1} \operatorname {AiryAi}\left (1, -\left (-8 i\right )^{{1}/{3}} x \right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBi}\left (1, -\left (-8 i\right )^{{1}/{3}} x \right )-2 x^{2} \left (\operatorname {AiryAi}\left (-\left (-8 i\right )^{{1}/{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (-\left (-8 i\right )^{{1}/{3}} x \right )\right )\right )}}{2 \operatorname {AiryAi}\left (-\left (-8 i\right )^{{1}/{3}} x \right ) c_{1} +2 \operatorname {AiryBi}\left (-\left (-8 i\right )^{{1}/{3}} x \right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 5.805 (sec). Leaf size: 413
DSolve[D[y[x],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 (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right ) \left (-2 x^2 \left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBiPrime}\left (2 (-1)^{5/6} x\right )+\left (1+i \sqrt {3}\right ) c_1 \operatorname {AiryAiPrime}\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right )} \\
y(x)\to \frac {\sqrt {\left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right ) \left (-2 x^2 \left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right )+\left (1+i \sqrt {3}\right ) \operatorname {AiryBiPrime}\left (2 (-1)^{5/6} x\right )+\left (1+i \sqrt {3}\right ) c_1 \operatorname {AiryAiPrime}\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \left (\operatorname {AiryBi}\left (2 (-1)^{5/6} x\right )+c_1 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )\right )} \\
y(x)\to -\frac {\sqrt {-\operatorname {AiryAi}\left (2 (-1)^{5/6} x\right ) \left (2 x^2 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )+\left (-1-i \sqrt {3}\right ) \operatorname {AiryAiPrime}\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )} \\
y(x)\to \frac {\sqrt {-\operatorname {AiryAi}\left (2 (-1)^{5/6} x\right ) \left (2 x^2 \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )+\left (-1-i \sqrt {3}\right ) \operatorname {AiryAiPrime}\left (2 (-1)^{5/6} x\right )\right )}}{\sqrt {2} \operatorname {AiryAi}\left (2 (-1)^{5/6} x\right )} \\
\end{align*}