Differential Equations with MATLAB Chapter 8

Using ode45 to find a vertical asymptote

Using ode45 to find a vertical asymptote

We look at the equation

$y' = x + y^2,\ y(0)=1$

What happens when we look for an exact solution?

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

              airy(1, -x) #1
airy(3, -x) - --------------
                    #2
----------------------------
              airy(0, -x) #1
airy(2, -x) - --------------
                    #2

where

               2/3                  / 2 \2
   #1 == 2 pi 3    - 3 sqrt(3) gamma| - |
                                    \ 3 /

               1/6          / 2 \2
   #2 == 2 pi 3    + 3 gamma| - |
                            \ 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.

Differential Equations with MATLAB Chapter 8

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Using ode45 to find a vertical asymptote