1. Section 6.1
Slope Fields

2. Differential Equations
An equation like 𝑑𝑦 𝑑𝑥 =𝑥 𝑒 𝑦 is called a differential equation because it contains a derivative.
If you find all of the functions y that satisfy the differential equation, then you have solved the differential equation.

3. Slope field
A slope field for the first order differential equation 𝑑𝑦 𝑑𝑥 =𝑓 𝑥, 𝑦 is a plot of short line segments with slopes f(x, y) for a lattice of points (x, y) in the plane.
A slope field can give you a general idea of what the solution to the differential equation looks like.

4. Indefinite integrals
Depending on where you place your pencil and begin drawing, the slope field can provide many different graphs that satisfy a particular differential equation.
We have seen this before with indefinite integrals.
The indefinite integral is the set of all antiderivatives to a function f(x):
𝑓 𝑥 𝑑𝑥=𝐹 𝑥 +𝐶

5. Find the following indefinite integrals… 1. 4𝑥 3 dx
Examples
Find the following indefinite integrals…
𝑥 3 dx
𝑥 𝑥 − cos 𝑥 dx
Adding a constant does not change the derivative because it does not affect the value of the slope at a given value x.

6. Initial value problems
Often the goal is to find a particular equation f(x) that both satisfies the differential equation and a given initial condition.
The initial condition is a value of f for one value of x. Graphically, it gives you a place to start in your slope field. Analytically it allows you to solve for the value of the constant in your indefinite integral.

7. Example
Solve the initial value problem….
𝑑𝑦 𝑑𝑥 = 6𝑥 2 −12𝑥+7, 𝑦 −2 =8

8. Initial value examples
Solve the initial value problem.
1. 𝑓 ′′ 𝑥 = 𝑥 −3/2 , 𝑓 ′ 4 =2, 𝑓 0 =0.
2. 𝑓 ′′ 𝑥 = sin 𝑥 , 𝑓 ′ 0 =1, 𝑓 0 =6.

9. Physics application
A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet.
(a) Find the position function giving the height h as a function of time t.
(b) When does the ball hit the ground?
(-32 ft/sec2 is the acceleration due to gravity.)