Differential Equations

I.  Homework

II.  Differential Equations

A differential equation is an equation that involves derivatives.  Examples are

y'' + 3y - 2 = 0

x²y' - 3xy² = 1/x

dy/dx - 2e^x = y  

III.  Solutions to Differential Equations

We say that y = f(x) is a solution to a given differential equation if substitution yields a true statement.

Example

Verify that y = 2e^2x is a solution to the differential euqation y'' - y' - 2y = 0.

Solution:

We find:  y' = 4e^2x, y'' = 8e^2x

Substituting into the differential equation:

8e^2x - 4e^2x - 2e^2x = 0

Exercise:

Show that y = 3e^-x is also a solution to the differnetial euqation.

IV.  Particular Solutions

Example:  An object in motion follows the differential equation

y'' + 3y' - 4y = 0

A.  Show that a general solution to this differntial eqation is

y = ce^x + ke^-4x

B.  If y(0) = 10 and y'(0) = 4 find the particular solution.

Solution:

A.  We have

y' = ce^x - 4ke^-x and y'' = ce^x + 16ke^-x

substituting we get

ce^x + 16ke^-x  + 3(ce^x - 4ke^-x) - 4(ce^x + ke^-4x) = 0

B.  y(0) = 3 gives

c + k = 5

and y'(0) = 4 gives

c - 4k = 0

Subtracting the equations gives

5k = 10 or k = 2 hence c = 3.  The particular solution is

y = 2e^x + 3e^-4x