2.27.2 Problem 2

Internal problem ID [13663]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2 Equations Containing Power Functions. page 213
Problem number : 2
Date solved : Thursday, January 01, 2026 at 02:15:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

2.27.2.1 second order airy

0.214 (sec)

\begin{align*} y^{\prime \prime }-\left (a x +b \right ) y&=0 \\ \end{align*}

Entering second order Airy solverThis is Airy ODE. It has the general form

\[ a y^{\prime \prime } + b y^{\prime } + c x y = F(x) \]

Where in this case

\begin{align*} a &= 1\\ b &= 0\\ c &= \frac {-a x -b}{x}\\ F &= 0 \end{align*}

Therefore the solution to the homogeneous Airy ODE becomes

\[ y = c_1 \operatorname {AiryAi}\left (\frac {\left (-a x -b \right ) \left (-a \right )^{{1}/{3}}}{a}\right )+c_2 \operatorname {AiryBi}\left (\frac {\left (-a x -b \right ) \left (-a \right )^{{1}/{3}}}{a}\right ) \]

Summary of solutions found

\begin{align*} y &= c_1 \operatorname {AiryAi}\left (\frac {\left (-a x -b \right ) \left (-a \right )^{{1}/{3}}}{a}\right )+c_2 \operatorname {AiryBi}\left (\frac {\left (-a x -b \right ) \left (-a \right )^{{1}/{3}}}{a}\right ) \\ \end{align*}

2.27.2.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)-(a*x+b)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {AiryAi}\left (\frac {a x +b}{\left (-a \right )^{{2}/{3}}}\right )+c_2 \operatorname {AiryBi}\left (\frac {a x +b}{\left (-a \right )^{{2}/{3}}}\right ) \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a Liouvillian solution using Kovacics algorithm 
<- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   <- Bessel successful 
<- special function solution successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )-\left (a x +b \right ) y \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\left (a x +b \right ) y \left (x \right ) \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (x \right )\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )+\left (-a x -b \right ) y \left (x \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y \left (x \right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y \left (x \right )=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )}{\sum }}a_{k} x^{k +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y \left (x \right )=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )+m}{\sum }}a_{k -m} x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d^{2}}{d x^{2}}y \left (x \right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\moverset {\infty }{\munderset {k =2}{\sum }}a_{k} k \left (k -1\right ) x^{k -2} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +2} \left (k +2\right ) \left (k +1\right ) x^{k} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & -a_{0} b +2 a_{2}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k +2} \left (k +2\right ) \left (k +1\right )-a_{k} b -a_{k -1} a \right ) x^{k}\right )=0 \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & -a_{0} b +2 a_{2}=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{2}+3 k +2\right ) a_{k +2}-a_{k -1} a -a_{k} b =0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \left (\left (k +1\right )^{2}+3 k +5\right ) a_{k +3}-a_{k} a -a_{k +1} b =0 \\ \bullet & {} & \textrm {Recursion relation that defines the series solution to the ODE}\hspace {3pt} \\ {} & {} & \left [y \left (x \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +3}=\frac {a_{k} a +a_{k +1} b}{k^{2}+5 k +6}, -a_{0} b +2 a_{2}=0\right ] \end {array} \]

2.27.2.3 ✓ Mathematica. Time used: 0.011 (sec). Leaf size: 36

ode=D[y[x],{x,2}]-(a*x+b)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \operatorname {AiryAi}\left (\frac {b+a x}{a^{2/3}}\right )+c_2 \operatorname {AiryBi}\left (\frac {b+a x}{a^{2/3}}\right ) \end{align*}

2.27.2.4 ✓ Sympy. Time used: 0.045 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq((-a*x - b)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} Ai\left (\sqrt [3]{a} x + \frac {b}{a^{\frac {2}{3}}}\right ) + C_{2} Bi\left (\sqrt [3]{a} x + \frac {b}{a^{\frac {2}{3}}}\right ) \]