AP Calc Notes: DEM – 8 Solving DIFFerential EQuations (“Diffy-Q’s”)
Ex: Solve for y if
dy
= 2 x and y(2) = 6
dx
1) separate the variables (tip: keep multiplied
constants with x)
2) integrate each side
3) + C to one side (why?)
4) use initial condition to find value for C
5) solve for y [recall: solution is a continuous
function]
Ex: Solve for f ( x ) if f ' ( x ) =
3
and f(1) = -10
y
1) separate the variables (tip: keep multiplied
constants with x)
2) integrate each side
3) + C to one side (why?)
4) use initial condition to find value for C
5) solve for y [recall: solution is a continuous
function]
Ex: Solve for y if y ' = 2 y and the curve passes through (0, 4)
1) separate the variables (tip: keep multiplied
constants with x)
2) integrate each side
3) + C to one side (why?)
Change order of find constant and solve for
exponential functions
4) solve for y
5) use initial condition to find value for A
Writing a differential equation from a word problem:
The rate of change of a population is directly proportional to the population.
The rate of change of a population is indirectly (or inversely) proportional to the population squared.
Ex: The rate of change of y is directly proportional to 1 – 2y with a constant of proportionality of 3. Find the
equation for y if the curve passes through the point (1, 4).