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Slide 6- 1
What you’ll learn about• Differential Equations• Slope Fields• Euler’s Method
… and whyDifferential equations have been a primemotivation for the study of calculus and remainso to this day.
First, a little review:
Consider:
then:or
It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears.
However, when we try to reverse the operation:
Given: find We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.
If we have some more information we can find C.
Given: and when , find the equation for .
This is called an initial value problem. We need the initial values to find the constant.
An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.
Slide 6- 4
Differential EquationAn equation involving a derivative is called a
differential equation. The order of a
differential equation is the order of the highest
derivative involved in the equation.
Slide 6- 5
Example Solving a Differential Equation
2Find all functions that satisfy 3 cos .
dyy x x
dx
2
3
The solution can be any antiderivative of 3 cos , which can be any function of the form sin .
x xy x x C
Slide 6- 6
First-order Differential Equation
If the general solution to a first-order differential
equation is continuous, the only additional
information needed to find a unique solution is
the value of the function at a single point, called
an initial condition. A differential equation
with an initial condition is called an initial-value
problem. It has a unique solution, called the
particular solution to the differential equation.
Slide 6- 7
Example Solving an Initial Value Problem
2Find the particular solution to the equation 3
1whose graph passes through the point 1, .
2
xdye x
dx
2 2
2
2
2 2 2
1 3The general solution is .
2 2Applying the initial condition, we have
1 1 3, from which we conclude that
2 2 21
2 . Therefore, the particular equation is2
1 3 12 .
2 2 2
x
x
y e x C
e C
C e
y e x e
Slide 6- 8
Example Using the Fundamental Theorem to Solve an Initial Value Problem
2
Find the solution to the differential equation '( ) cos for which (3) 5.f x x f
2
3The solution is ( ) cos 5.xf x t dt
Initial value problems and differential equations can be illustrated with a slope field.
Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but a recent AP test asked students to draw a simple one by hand.
Draw a segment with slope of 2.
Draw a segment with slope of 0.
Draw a segment with slope of 4.
0 0 00 1 00 00 0
23
1 0 21 1 2
2 0 4
-1 0 -2
-2 0 -4
If you know an initial condition, such as (1,-2), you can sketch the curve.
By following the slope field, you get a rough picture of what the curve looks like.
In this case, it is a parabola.
Integrals such as are called definite integrals
because we can find a definite value for the answer.
The constant always cancels when finding a definite integral, so we leave it out!
Integrals such as are called indefinite integrals
because we can not find a definite value for the answer.
When finding indefinite integrals, we always include the “plus C”.