87.10.1 problem 1

Internal problem ID [23402]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 79
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:41:06 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+y x&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)+x*y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\Gamma \left (\frac {2}{3}\right ) \left (3^{{2}/{3}} \operatorname {AiryAi}\left (-x \right )+3^{{1}/{6}} \operatorname {AiryBi}\left (-x \right )\right )}{2} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 41

ode=D[y[x],{x,2}]+x*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \sqrt [6]{3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (\sqrt {3} \operatorname {AiryAi}\left (\sqrt [3]{-1} x\right )+\operatorname {AiryBi}\left (\sqrt [3]{-1} x\right )\right ) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {3^{\frac {2}{3}} Ai\left (- x\right ) \Gamma \left (\frac {2}{3}\right )}{2} + \frac {\sqrt [6]{3} Bi\left (- x\right ) \Gamma \left (\frac {2}{3}\right )}{2} \]