Separation of Variables Solving First Order Differential Equations.

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Presentation transcript:

Separation of Variables Solving First Order Differential Equations

Solving ODEs What is Solving an ODE? Eliminating All Derivatives Explicit Form Implicit Form

This Chapter 1st Order (Only First Derivative) Linear and Nonlinear

Calculus Brain Teaser: ?

Today We will try to make problems look like:

Why? Want to “Get Rid of” This Derivative

Why? So we integrate the left side Have to integrate right side too

Separation of Variables No more derivatives! Implicit (General) Solution

Separation of Variables No more derivatives! Implicit (Specific) Solution If we havecan solve for C

Chain Rule Remember, y is a function of t

Chain Rule

So To Solve Think of it as: (Reversing the Chain Rule)

So To Solve Think of it as: Find by solving Keep equation balanced by solving

The whole process… For an equation of the form: (May need to manipulate equation to get here)

The whole process… For an equation of the form: Separate the variables

The whole process… For an equation of the form: Separate the variables is

The whole process… For an equation of the form: Separate the variables Integrate both sides Perhaps solve for y, or C (if initial condition)

A Simple Example

A Convenient Technique

“Cross Multiply”

A Convenient Technique

Integral Curves Is solved by: or Equation for an ellipse (for different values of C)

Integral Curves Plots of Solutions for Different Values of -C are called “Integral Curves” Integral Curves Show Different Behaviors for Different Initial Conditions

Integral Curves

In Summary To Solve an ODE, eliminate derivatives One method for first order linear/nonlinear ODES Separation of Variables (Reverse Chain Rule) Integral curves are solution curves for different values of C

Questions?