1. 7-1 Differential equations & Slope fields

2. Def: Differential Equation: an equation that involves a derivative
The order of a differential equation is order of highest deriv. involved
Ex 1) Find all functions y that satisfy
Solution is antiderivative
 any form of:
general solution
*What if we are looking for a particular solution?
Need an initial condition to solve for C

3. Ex 2) Find the particular solution to the equation
whose graph passes through the point (1, 0).
General solution:
Ex 3) (Handling Discontinuity in an Initial Value Problem)
Find the particular solution to the equation
whose graph passes through (0, 3).

4. Ex 4) (Using Fund Thm to Solve an Initial Value Prob)
Find the solution to the differential equation
for which f (7) = 3.
Check:
Works! 
Ex 5) Graph the family of functions that solve the
differential equation
y = sin x + C
Graph these  {–3, –2, –1, 0, 1, 2, 3}  L1
Y1 = sin x + L1  Zoom TRIG

5. Ex 6) Construct a slope field for the differential equation
Slope Fields
What if we want to reproduce a family of graphs without solving a differential equation? (Which is sometimes really hard to do.) We can, by examining really small pieces of the “slopes” of the differential curve and repeating this process many times over. The result is called a slope field.
Ex 6) Construct a slope field for the differential equation
Think about y values at certain x
value spots…
When x = 0, cos x = 1
When x =  or x = –, cos x = –1
When x = (odd mult) , cos x = 0
When x = 2 or x = –2, cos x = 1 2
–2 4 –4

6. *Note: You can graph the general solution with a slope field even if you cannot find it analytically

7. Ex 7) Use a calculator to construct a slope field for the
differential equation and sketch a graph of the
particular solution that passes through the point (2, 0).
Slopes = 0 along x + y = 0
Slopes = –1 along x + y = –1
Slopes get steeper as x inc
Slopes get steeper as y inc
Draw a smooth curve thru point (2, 0) that follows slopes in slope field

8. Horizontal when x = 0 & Vertical when y = 0
Ex 8) Use slope analysis to match each of the following differential equations with one of the slope fields pictured.
Zero slope along
x – y = 0 B
Zero slope along
both axes D
Horizontal when x = 0 & Vertical when y = 0 A
Vertical when x = 0
& Horizontal when y = 0 C

9. homework
Pg. 331 #1 – 46 (skip mult of 3), 50, 59