Systems of First Order Differential Equations

1. An mth-order System:
   

2. Lipschitz condition [several variables]:
   

   

3. Existence and Uniqueness:
   

Methods to Solve Systems of Equations

1. What?

   Methods to solve systems of first-order differential equations are generalizations of the methods for a single first-order equation presented earlier in this chapter.

2. Generalizing Runge-Kutta Order-Four:
   Show Derivation

3. Algorithm:
   Algorithm

4. Example [Kirchoffs Law]:
   Illustration

Higher-Order Differential Equations

1. What?

   New techniques are not required for solving initial-value problems whose equations have orders higher than one.

   By relabeling the variables, we can reduce a higher-order differential equation into a system of first-order differential equations and then apply one of the methods we have already discussed.

2. Converting an mth-order IVP to 1st-order System of Equations:
   

3. Example:
   Example