AP Calc AB
Sec 6.1 day 1 Differential Equations
Read p.404-405
We will cover goal #1 today: Use initial conditions to find particular solutions of differential equations.
Note: A Differential Equation is simply an equation with a derivative in it. A solution of the
differential equation is not a number, but an equation or function.
A function is a solution of a differential equation if the equation is satisfied, meaning it is true, when y
and its derivatives are substituted into the equation.
2
2 -2x
1: Show that is a solution of the diffe
rential equation ' 2 0.
First find all of the derivatives that are used in the differential equation.
if , then y'=-2e . Substitute both of th
x
x
Ex y e y y
y e
−
−
= + =
=
-2x 2
2
ese into the differential eq.
' 2 0
-2e 2( ) 0
0 = 0
What if 5
x
x
y y
e
y e
−
−
+ =
+ =
=-2x -2x
-2x 2
instead? y'=5(-2)e or y' = -10e .
' 2 0
-10e 2(5 ) 0
0 = 0
I
x
y y
e−
+ =
+ =
2
2
n conclusion, every solution of ' 2 0 is in the form
where C is an y real number. is called the .
x
x
y y y Ce
y Ce general solution
−
−
+ = =
=
Ex 1 involves a “1st order” differential equation because y’ is the highest-order derivative in the
equation and the general solution involves 1 constant, C. A “2nd order” differential equation involves
y” and possibly y’ and y and its general solution involves 2 arbitrary constants. It can be shown that a
differential equation of the ”nth order” has a general solution involving n arbitrary constants.
There are more examples on p.404.
2x
If you choose values for the arbitrary c
onstants and graph them,
you will create a family of curves known as .
For ex1, the general solution is y=Ce .
Choosing arbitrary values
for C
solution curves
2x 2x 2x
2x 2x 2x
gives us a family of curves like y=12e ,
y=5e , y=8e ,
y=-2e , y=-4e , y=-7e .
