Separation of Variables Solving First Order Differential Equations

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:

?

Calculus Brain Teaser:

TodayWe 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 have can solve for C

Chain Rule

Remember, y is a

function of t

Chain Rule

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 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 Simple Example

A Simple Example

A Simple Example

A Simple Example

A Simple Example

A Simple Example

A Convenient Technique

A Convenient Technique

A Convenient Technique

A Convenient Technique

“Cross Multiply”

A Convenient Technique

A Convenient Technique

A Convenient Technique

Integral CurvesIs 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

Integral Curves

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?