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.

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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.

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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

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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.

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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

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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”.