1. Differential Equations
Sec.1: Slope Fields

2. Definition: Differential Equation
A differential equation is an equation that involves derivatives.
ex: y’+2y=0, x2y’’–3xy’+3y=0
A function y=f(x) is called a solution to a differential equation if f(x) and its derivatives satisfy the differential equation.

3. Ex1: Verifying Solutions
Determine whether the function is a solution to the differential equation y’’–y=0.
y=sin(x)
y=4e-x
y=Cex, where C is any constant

4. General Solution Is y=C/x a solution to xy’+y=0? Yes
In y=C/x, C is any real number. So y is a general solution to the differential equation.
A particular solution can be obtained from an initial condition.

5. Ex2: Particular Solution
Consider the differential equation xy’–3y=0.
Verify that y=Cx3 is a solution.
y’=3Cx2
xy’–3y = x(3Cx2)–3(Cx3) = 0
Find the particular solution determined by the initial condition that y=2 when x=–3
y=Cx3, 2=C(-3)3, C=-2/27
y=-2x3/27

6. Slope Fields
A slope field of a differential equation y’=F(x,y) is a set of short line segments with slope F(x,y) at selected points.

7. Ex 3: Sketching a Slope Field
Sketch a slope field for the differential equation y’=x–y for points (-1,1), (0,1), and (1,1)
Slope at (-1,1) is -2
Slope at (0,1) is -1
Slope at (1,1) is 0

8. Ex4: Identifying Slope Fields

9. Ex5: Sketching a Solution
Sketch a slope field for the differential equation y’=2x+y.
Use the slope field to sketch the solution that passes through (1,1).

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