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AP CALCULUS NOTES
SECTION 9.1 FIRST-ORDER DIFFERENTIAL EQUATIONS AND APPLICATIONS
Recall from Sec. 6.2:
Differential Equation – an equation that involves a derivative of an unknown function.
dy
 f  x  is called a differential equation
dx
 The order of a differential equation is the order of the highest derivative that it
contains.
General First-Order Differential Equations and Solutions:
dy
 f  x, y  in which f  x, y  is a
A.) A first-order differential equation is an equation
dx
function of two variables defined on a region in the xy-plane.
 A function y  y  x  is a solution of a differential equation on an open interval I if the
equation is satisfied when y and its derivative are substituted.
Ex.1.) Verifying Solution Functions
C
Show that every member of the family of functions y   2 is a solution of the first-order
x
dy 1
  2  y  on the interval  0,   , where C is any constant.
differential equation
dx x
1.) The general solution to a first-order differential equation is a solution that contains all
possible solutions (contains an arbitrary constant).
2.) The particular solution satisfies the initial condition y  x0   y0 . Thus, a first-order initial
value problem is a differential equation
dy
 f  x, y  whose solution must satisfy an initial
dx
condition.
Ex.2.) Verifying that a Function is a Particular Solution.
1
Show that the function y   x  1  e x is a solution to the first-order initial value problem
3
dy
2
 y  x where y  0   .
dx
3
B.) A first-order separable differential equation is one that can be written in the form
dy
h  y   g  x  , and it involves the integration of both x and y.
dx
METHOD OF SEPARATION OF VARIABLES:
1.) Collect all y terms with dy and all x terms with dx .
2.) Integrate both sides.


h  y  dy  g  x  dx
 h  y  dy   g  x  dx
3.) In some cases, the solution y will be defined implicitly as a function of x.
dy
 1  y 2  e x .
Ex.3.) Solve the differential equation
dx
Ex.4.) Solve the differential equation
dy
6x2
.

dx 2 y  cos y
dy x 2
. Then find the solution of this equation that

dx y 2
equation that satisfies the initial condition y  0  2 .
Ex.5.) Solve the differential equation
Ex.6.) Solve the equation y  x 2 y .
Not every first-order differential equation is separable.
C.) A first-order linear differential equation is one that can be written in the form
dy
 P  x y  Q  x .
dx
For these we use the METHOD OF INTEGRATING FACTORS:
P x  dx
1.) Calculate   e
. This is called the integrating factor.
dy
d
 P  x  y  Q  x  by  and express the result as
  y   Q  x  .
dx
dx
3.) Integrate both sides of the equation obtained in Step 2 and then solve for y. Be sure to
include a constant of integration in this step.
dy
 3x 2 y  6 x 2 .
Ex.7.) Solve the differential equation
dx
2.) Multiply both sides of
Ex.8.) Solve the differential equation x
dy
 x 2  3 y, x  0 .
dx
Ex.9.) Solve the equation xy  x 2  3 y, x  0 given the initial condition y 1  2 .
Ex.10.) Find the solution of the initial value problem: x 2 y  xy  1, x  0 , y 1  2 .