Differential Equations with MATLAB Chapter 8

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.