Introduction to Differential Equations Simplified Study notes (Part 2), Study notes of Differential Equations

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INTRODUCTION TO

DIFFERENTIAL

EQUATIONS

The Basics Explained (Part 2)

One of the goals of this course is to SOLVE differential equations. This presentation will introduce the concept of a solution of an ODE.

Solution of an ODE

Example: is a solution to the first-order ODE

Substitute the derivative of the solution

into the DE to see if each side of the

equation is the same for every x on the

interval.

Verification of a Solution

Function Vs Solution: Function: (^) Solutio n:

• (^) The graph of a solution of an ODE is called a solution

curve.

• (^) The graph of a function may differ from its solution curve.

Implicit and Explicit Solutions: Explicit Solution Any solution that is given in the form y = f(x). Example: Explicit solution where c is an arbitrary constant Implicit solution: Any solution that is not in the form of an explicit solution. Example: Implicit solution

One-parameter family It has a one-parameter family explicit solution: This is a first-order linear ODE on the open interval (-∞,∞) c is an arbitrary parameter

Two-parameter family It has a two-parameter family explicit solution: This is a second-order linear ODE on the open interval (-∞,∞) c1 and c2 are arbitrary parameters

A solution of an equation that does not have arbitrary parameters/constants is called a PARTICULAR SOLUTION. Families of Solutions:

Notes in this presentation was taken from the text book, Dennis G. Zill, Warren S Wright- Differential Equations with Boundary-Value Problems- Brooks Cole (2012)