# Differential Equations with MATLAB Chapter 8

## Contents

## Using ode45 to find a vertical asymptote

We look at the equation

What happens when we look for an exact solution?

syms x y = dsolve('Dy = x + y^2, y(0)=1', 'x'); pretty(y)

airyAi(-x, 1) #1 airyBi(-x, 1) - ---------------- #2 -------------------------------- airyAi(-x, 0) #1 airyBi(-x, 0) - ---------------- #2 where 2/3 1/2 2 #1 = 2 pi 3 - 3 3 gamma(2/3) 1/6 2 #2 = 2 pi 3 + 3 gamma(2/3)

The solution invoves two types of Airy functions. It isn't very easy to see what it means. What happens when we plot the numeric solution obtained using **ode45**?

f = @(x,y) x+y^2; [t,ya] = ode45(f,[0,1],1); plot(t,ya)

Warning: Failure at t=9.305611e-01. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.776357e-15) at time t.

It has a vertical asymptote between 0.9 and 0.95. We'll plot it on some smaller intervals.

[t,ya] = ode45(f,[0,0.9],1); plot(t,ya)

[t,ya] = ode45(f,[0,0.95],1); plot(t,ya)

Warning: Failure at t=9.305618e-01. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.776357e-15) at time t.

[t,ya] = ode45(f,[0,0.93],1); plot(t,ya)

[t,ya] = ode45(f,[0,0.94],1); plot(t,ya)

Warning: Failure at t=9.305485e-01. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.776357e-15) at time t.

The asymptote is at about t=0.93. You can zoom in to get a better idea.