01 Differential Equations.notebook
April 23, 2014
Differential Equations
A differential equation is an equation that deals with an "x", a "y" and derivatives of "y".
Solutions to Differential Equations
A General Solution to a differential equation would x
be written in a generalized form such as: y = Ce
A Specific Solution to a differential equation would x be written in a specific form such as: y = 4e
(in this case, "C" is specified as 4)
Differential Equations
Verifying Solutions to Differential Equations
Determine whether the function is a solution of the differential equation:
∴ No!
∴ Yes!
Solutions?
1
01 Differential Equations.notebook
April 23, 2014
Verifying Solutions to Differential Equations
Is x2 + y2 = cy a solution of What can I do to verify this???
Take the derivative of x2 + y2 = cy
OR
Take the integral of y'
This is not the same, but can we maybe manipulate it to get it in the form asked for?
Verifying a Solution
Verifying Solutions to Differential Equations
Is y = x 2 ex a solution of xy' ­ 2y = x 3 ex?
So, plug y' into equation and see if it works
Yes!!
Verifying #2
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01 Differential Equations.notebook
April 23, 2014
Family of Equations
For xy' ­ 3y = 0, y = Cx3 is a (general) solution
What would the graphs of y look like?
Use the Graphing calculator to check out the graphs as you change the values of "C". Pick positive & negative values.
Family of Equations
Family of Equations
For xy' ­ 3y = 0, y = Cx3 is a (general) solution
Find the particular solution given the initial condition that y = 2 when x = ­3?
Plugging in 2 for "y" and -3 for "x":
Family of Equations
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01 Differential Equations.notebook
April 23, 2014
Slope Fields:
A slope field shows the general shape of ALL the solutions and is an visual aid for the directions of the solutions to a differential equation.
Slope Fields
Match the Slope Fields with the
Differential Equation!
Matching Practice
4
01 Differential Equations.notebook
April 23, 2014
Match the Slope Fields with the
Differential Equation!
Matching Practice
Homework:
p411 (#1, 4, 5, 7, 9, 12, 13, 17, 21, 24,
28, 29, 30, 37, 41, 47, 57-60)
Homework
5
Attachments
TI­SmartView_TI­84.exe