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?