Calculus Section 6.1 Slope Fields
-Use initial conditions to find particular solutions of differential equations
-Use slope fields to approximate solutions of differential equations
Homework: page 409 #’s 1, 5, 13, 15,
19, 21, 25, 27, graph 57, 58
Every day, physical phenomena can be represented and described by differential equations. Things like
population growth, sales predictions, radioactive decay, and Newton’s Law of Cooling are examples that can
be described using differential equations.
A function y = f(x) is considered a ______________ to a differential equation if you can ___________________
the values of y into the differential equation and have the equation be _____________. These solutions are
called ___________________________ because they will work for any general constant, C, value.
Example Verifying solutions
Determine whether the following functions are solutions of the differential equation y ’’ – y = 0.
a) y = sinx
b) y = 4e-x
c) y = Cex
The name of a differential equation comes from the level of the derivative that appears in the equation. For
example, the equation y ‘ – xy = 0 is a ________________________________________________________.
The equation y ‘ +2y ‘’ = x is a ______________________________________________. The highest level of
derivative is the determining factor in naming the differential equation.
Graphically, the general solution of a first-order differential equation represents the family of curves known as
the __________________________. These represent the differential equation for each value assigned to the
arbitrary constant C.
Example
Consider the differential equation xy’ + y = 0. Confirm that
y = C/x is a general solution, and then graph the solution
curves for the function.
Finding a Particular Solution
Finding the particular solution to a differential equation can be found using ______________ conditions
similar to what we did with applications of derivatives.
Example
For the first-order differential equation xy’ – 3y = 0, verify that y = Cx3 is a solution. Then, find the particular
solution determined by the initial condition y = 2 when x = -3.
In order to find a particular solution, the number of initial conditions must match the number of
_______________________ in the general solution.
Slope Fields
A graphical approach to solving a differential equation is called a ______________________. These fields use
short segments to represent the slope of the differential equation at points throughout the coordinate plane.
The slope field displays the basic shape of all the general solutions to the differential equation.
Match the slope fields with their differential functions.
Sketch a slope field for the differential equation y’ = 2x + y.
Use the slope field to sketch the solution that passes through (1, 1).
a) y’ = x
b) y’ = y
c) y’ = x+y