Section 19.6

Intervals of Validity

Knowing a solution formula isn't enough. We need to know where it actually works.

Introduction

For Linear Equations, we can find the interval of validity just by looking at the coefficients.

For Non-Linear Equations, the interval of validity depends on the Initial Condition itself. A solution might "blow up" to infinity in finite time!

Theorems

Linear Theorem

For :

The unique solution exists on any interval containing where both and are continuous.

Non-Linear Theorem

For :

A unique solution exists near if both and are continuous there.
Catch: We can't predict the interval size without solving it.

Visual: Finite Blow-up

Interactive: Solution approaches vertical asymptote

For , changing the initial value changes where the solution explodes. This never happens in Linear equations.

Worked Examples

Example 1: Linear Interval

Consider , .

1. Standard Form:

2. Discontinuities:

breaks at .

requires and breaks at .

So bad points are .

3. Interval containing :

We are separated from 1 and must be positive.

Valid Interval: .

Example 2: Non-Linear Interval

Solve , and find interval of validity.

1. Solve:

2. Apply IC:

3. Interval:

Undefined at . Since , valid interval is .

Example 3: Discontinuous Coefficients

Find interval of validity for , .

1. Coefficients:

Discontinuous when , i.e., .

is continuous everywhere.

2. Plot Discontinuities:

Bad points:

3. Check Initial Condition:

. This lies between and .

4. Interval: