Existence and Uniqueness of Systems of Differential Equations

To prove that a linear system of differential equations are exist and have unique solutions on an interval first decompose the system into the from:

You'll notice that this is a similar form specified in the definition of differential equation.

Once the system is in this form you can #incomplete

Theorem 4.1 - Linear Case

Given a linear system of differential equations and both and have continuous components (continuity in ) then the IVP with has a unique solution all over .

Theorem 6.1 - Nonlinear and Linear Cases

Elementary Differential Equations - Kohler & Johnson - Second Edition - Pg. 395

Consider the initial value problem:

where the initial value point lies in the region defined by the inequalities:

Let and the partial derivatives given by the jacobian matrix be continuous in . Then the initial value problem has a unique solution that exists on some t-interval containing .

is continuous for .
is continuous in and .

Where is the jacobian matrix.

Second Order Linear IVP

1-Second Order Scalar DE (Sect 3-1 to 3-5) - make sure theorem below is correct

Let , , and be continuous functions on the interval , and let be in . Then the initial value problem

has a unique solution defined on the entire interval .