section 6.2 – differential equations (growth and decay)
TRANSCRIPT
Section 6.2 – Differential Equations (Growth and Decay)
Reminder: Directly Proportional
Two quantities are said to be in direct proportion (or directly proportional, or simply proportional), if one is a constant
multiple of the other. For example, y is proportional to x if k is a constant and:
y k x
Reminder: Inversely Proportional
Two quantities are said to be in inverse proportion if their product is constant. ( In other words, when
one variable increases the other decreases in proportion so that the product is unchanged.) For
example, if y is inversely proportional to x and if k is a constant:
ky x k or y
x
Solving Differential Equations
We are familiar with the simplest type of differential equations, namely y' = f(x). A solution is simply an antiderivative. For example:
Other differential equations can be in terms of x and y. For example:
2sec 2 5dydx x x
' 3y x y
Separable Equations
A more general class of first-order differential equations that can be solved directly by integration are the separable equations, which have the form:
dydx f x g y
The first derivative is the product
of…
a function in terms of
x…
and a function in terms of y.
Examples
Which first order differential equations below are separable?
21. sin
2. 3
3.
dydx
dydx
dydx
x y
xy y
x y
Separable because it is a product of a function
of x (sinx) and a function of y (y2)
3 1y x Separable because it is a
product of a function of x (3x+1) and a function of y (y)
Not separable because it can not be written as product of function of x and a function of y
Separation of Variables
If a first order differential equation is separable, use the following solution method:
1. Make sure the differential equation is written as a product of a function of x and a function of y.
2. Move all of the y terms on one side and all of the x terms on the other; this includes the dx and dy.
3. Integrate both sides.
4. Solve for y (if possible).
Example 1
Find the general solution to: 0dydxy x
Is this a separable equation?0dy
dxy x dydxy x
1dydx x y
YES. The derivative can be written as a product of a
function of x and a function of y.
Now use separation of variables to find the general solution.
1dydx x y
y dy x dxy dy x dx
2 21 12 2y x C
2 2 2y x C 2y x C C is arbitrary, so there is no
difference between 2C and C.
A C on both sides would be
redundant.
Example 2
Solve the initial value problem: ' , 0 3y t y y
The derivative IS written as a product of a function of t and a function of y.
Now use separation of variables to find the general solution.
'y t y dydt t y dyy t dt
dyy t dt
212ln y t C
212t Cy e
Since C is arbitrary, ±eC represents an arbitrary nonzero number. We can replace it with C:
In this example, t = x.
212tCy e e
212ty Ce
Now use the initial condition to find the particular solution:
212 03 Ce
3 C 2123 ty e
Example 3
Solve the initial value problem: 24 cos 3 0, 0 0dy
dxy y x y Can the derivative be written as a product of
a function of t and a function of y?Yes. Now use separation of variables to
find the general solution.
24 cos 3 0dydxy y x
24 cos 3dydxy y x
234 cos
dy xdx y y
24 cos 3y y dy x dx
24 cos 3y y dy x dx 2 32 siny y x C
Now use the initial condition to find the particular solution:
2 32 0 sin 0 0 C 0 C
2 32 siny y x Not every equation can be solved for y.
Annual GrowthJason bought a limited edition Lenny Dykstra signed rookie card for $250. Jason knows the price of such an awesome card will increase by 4.3% per year compounded once a year.
How much will the card be worth after 0 years? 1 year? 2 years? 3 years? 4years? 5 years?
Year Card Value
0 $250
1 $260.75
2 $271.96
3 $283.66
4 $295.85
5 $308.58
250 0.043 260.75 0.043 271.96 0.043 283.66 0.043 295.85 0.043
The rate of change in the new output is
directly proportional to
the previous output.
What about if it is compounded continuously?
“Continuous” Growth and DecayIf something is growing or decaying continuously exponentially, then the following holds:
The rate of change in the output (dy/dx) is proportional to the output (y).
In a calculus equation, this statement becomes:dy
k ydt
We can use separation of variables to find the general solution for a growth or decay situation.
“Continuous” Growth and Decay
The derivative IS written as a product of a function of t and a function of y.
Now use separation of variables to find the general solution.
dydt k y dyy k dt
dyy k dt
ln y kt C kt Cy e
Since C is arbitrary, ±eC represents an arbitrary nonzero number. We can replace it with C:
C kty e e
kty Ce
Find the general solution to: dydt k y
“Continuous” Growth and Decay
If y is a differentiable function of t such that y>0 and y'=ky, for some constant k, then
C = Initial Value
k = Proportionality Constant
kty Ce
0 : 0 1:k Growth k Decay
“Continuous” Growth and Decay
If:
Then: ktdyk y or y Ce
dt
Your Choice: Remember how to derive the general solution from the differential equation OR
memorize the general solution.
The rate of change in the output (dy/dx) is proportional to the output (y).