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# Lecture 1

## by wraithxjmin on Apr 24, 2008

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## Lecture 1Presentation Transcript

• CHAPTER 2 Differential Equation
• Introduction Consider x as an independent variable and y as dependent variable. An equation that involves at least one derivative of y with respect to x, e.g. Is known as a differential equation or common differential equation .
• b) c) d) a) Example of Differential Equation
• Order is the highest derivative
• Degree is the highest power of the highest
• derivative
• Examples: a)
This DE has order 2 (the highest derivative appearing is the second derivative) and degree 1 (the power of the highest derivative is 1.) Order & Degree
• In this chapter we only deal with first order,
• first degree differential equations.
• A solution for a differential equation is a
• function whose elements and derivatives may be
• substituted into the differential equation. There
• are two types of solution for differential
• equations
• General solution – The general solution of a differential equation contains an arbitrary constant c .
• Particular solution - The particular solution of a differential equation contains a specified initial value and containing no constant.
Solutions
• Examples Of General Solution
• This is already in the required form, so we
• simply integrate:
• , c is constant
• Example First we must separate the variables: Multiply throughout by dx Divide throughout by y Divide throughout by x
• This gives us: We now integrate ,
• Example First we must separate the variables: Multiply throughout by dx Divide throughout by Divide throughout by
• This gives us:
We now integrate:
• Separable Variables and Integrating Factors
• Example
• Solve the differential equation
First we must separate the variables: Consider :
• By using substitution,
• Consider
• Let
and By using integration by part
• The General Solution :
• Example
• Solve the differential equation
• when x =0, y=5
• First we must separate the variables:
We now integrate:
• We now use the information which means
• at , to find c .
So the particular solution is: gives
• Exercise : Solve the initial value problem. Express the solution implicitly. a. b.