4.3.2.4 Airy ode \(y^{\prime \prime }\pm kxy=0\) or \(y^{\prime \prime }+by^{\prime }\pm kxy=0\)

ode internal name "second order airy"

Sometimes this  is written as \(y^{\prime \prime }\pm k^{2}xy=0\). But it is the same ode. The power on \(k\) is not important. So in this below will show for generic \(k^{n}\).

This table gives the patterns to use for solving Airy ode. This result uses this general form of Airy ode

\[ Ay^{\prime \prime }\pm By^{\prime }\pm k^{n}\left ( ax+b\right ) y=0 \]

Hence in this table, if \(y^{\prime }\) is missing, we just replace \(B=0\). This all assumes \(k,A,B,a,c\) do not depends on \(x\). The solution to the above is given by

\[ y=c_{1}e^{\left ( \frac {\mp Bx}{2A}\right ) }\operatorname {AiryAi}\left ( -\frac {\left ( A\left ( ax+b\right ) k^{n}-\frac {B^{2}}{4}\right ) \left ( \frac {\pm k^{n}a}{A}\right ) ^{\frac {1}{3}}}{aAk^{n}}\right ) +c_{2}e^{\left ( \frac {\mp Bx}{2A}\right ) }\operatorname {AiryBi}\left ( -\frac {\left ( A\left ( ax+b\right ) k^{n}-\frac {B^{2}}{4}\right ) \left ( \frac {\pm k^{n}a}{A}\right ) ^{\frac {1}{3}}}{aAk^{n}}\right ) \]

The only thing we need to watch for, is the sign on \(B\) and on \(k^{n}\). If the sign is negative in the ode, then we use \(e^{\left ( \frac {Bx}{2A}\right ) }\) and if the sign is positive on \(B\) then we use \(e^{\left ( -\frac {Bx}{2A}\right ) }\). For \(k^{n}\), the leading sign do not change in the solution.  Below are some examples

ODE	Values	solution
\(y^{\prime \prime }-k^{n}xy=0\)	\(A=1,B=0,a=1,b=0\)	\(y=c_{1}\operatorname {AiryAi}\left ( -\left ( -k^{n}\right ) ^{\frac {1}{3}}x\right ) +c_{2}\operatorname {AiryBi}\left ( -\left ( -k^{n}\right ) ^{\frac {1}{3}}x\right ) \)
\(y^{\prime \prime }+k^{n}xy=0\)	\(A=1,B=0,a=1,b=0\)	\(y=c_{1}\operatorname {AiryAi}\left ( -\left ( k^{n}\right ) ^{\frac {1}{3}}x\right ) +c_{2}\operatorname {AiryBi}\left ( -\left ( k^{n}\right ) ^{\frac {1}{3}}x\right ) \)
\(y^{\prime \prime }-k^{2}\left ( x+3\right ) y=0\)	\(A=1,B=0,a=1,b=3\)	\(y=c_{1}\operatorname {AiryAi}\left ( -\left ( -k^{2}\right ) ^{\frac {1}{3}}\left ( x+1\right ) \right ) +c_{2}\operatorname {AiryBi}\left ( -\left ( -k^{2}\right ) ^{\frac {1}{3}}\left ( x+1\right ) \right ) \)
\(5y^{\prime \prime }+2y^{\prime }-k^{4}\left ( 3x+4\right ) y=0\)	\(A=5,B=2,a=3,b=4\)	\(c_{1}e^{\left ( \frac {-x}{5}\right ) }\operatorname {AiryAi}\left ( -\frac {\left ( 5\left ( 3x+4\right ) k^{4}-1\right ) \left ( \frac {-k^{4}\left ( 3\right ) }{5}\right ) ^{\frac {1}{3}}}{15k^{4}}\right ) +c_{2}e^{\left ( \frac {-2x}{10}\right ) }\operatorname {AiryBi}\left ( -\frac {\left ( 5\left ( 3x+4\right ) k^{4}-1\right ) \left ( \frac {-k^{4}\left ( 3\right ) }{5}\right ) ^{\frac {1}{3}}}{15k^{4}}\right ) \)