Section 6.2 Differential Equations (Growth and Decay)

Size: px

Start display at page:

Download "Section 6.2 Differential Equations (Growth and Decay)"

Wilfrid Sharp
5 years ago
Views:

1 Section 6. Differential Equations (Growth and Deca)

2 Reminder: Directl Proportional Two quantities are said to be in direct proportion (or directl proportional, or simpl proportional), if one is a constant multiple of the other. For example, is proportional to x if k is a constant and: k x

3 Reminder: Inversel 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 is inversel proportional to x and if k is a constant: x k or k x

4 Solving Differential Equations We are familiar with the simplest tpe of differential equations, namel ' = f(x). A solution is simpl an antiderivative. For example: sec x x 5 Other differential equations can be in terms of x and. For example: ' x 3

5 Separable Equations A more general class of first-order differential equations that can be solved directl b integration are the separable equations, which have the form: The first derivative is the product of f x g a function in terms of x and a function in terms of.

6 Examples Which first order differential equations below are separable? 1. sin. 3x 3. x x Separable because it is a product of a function of x (sinx) and a function of ( ) 3x1 Separable because it is a product of a function of x (3x+1) and a function of () Not separable because it can not be written as product of function of x and a function of

7 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.. Move all of the terms on one side and all of the x terms on the other; this includes the and. 3. Integrate both sides. 4. Solve for (if possible).

8 Example 1 Find the general solution to: x0 Is this a separable equation? x 0 x x YES. The derivative can be written as a product of a function of x and a function of. 1 Now use separation of variables to find the general solution. A C on both sides would be redundant. C is arbitrar, so there is no difference between C and C. x x 1 x x C x C 1 1 x C

9 Example Solve the initial value problem: ' t, 0 3 Now use separation of variables to find the general solution. ' dt t t t dt t dt 1 ln t C 1 t C e C e e 1 t In this example, t = x. The derivative IS written as a product of a function of t and a function of. Since C is arbitrar, ±e C represents an arbitrar nonzero number. We can replace it with C: Ce 1 t Now use the initial condition to find the particular solution: 3 Ce C 1 t 3 e

10 Example 3 Solve the initial value problem: 4 cos 3x 0, cos 3x 0 4 cos 3x 3x 4 cos 4 cos 3x 4 cos 3x sin 3 x C Can the derivative be written as a product of a function of t and a function of? Yes. Now use separation of variables to find the general solution. Now use the initial condition to find the particular solution: 3 0 sin 0 0 C 0 C sin x 3 Not ever equation can be solved for.

11 Annual Growth Jason bought a limited edition Lenn Dkstra signed rookie card for $50. Jason knows the price of such an awesome card will increase b 4.3% per ear compounded once a ear. What about if it is How much will the card be worth after 0 ears? 1 ear? ears? 3 ears? 4ears? 5 ears? Year compounded continuousl? Card Value 0 $50 1 $60.75 $ $ $ $ The rate of change in the new output is directl proportional to the previous output.

12 Continuous Growth and Deca If something is growing or decaing continuousl exponentiall, then the following holds: The rate of change in the output (/) is proportional to the output (). In a calculus equation, this statement becomes: dt k We can use separation of variables to find the general solution for a growth or deca situation.

13 Continuous Growth and Deca Find the general solution to: dt k Now use separation of variables to find the general solution. dt k k dt k dt ln kt C kt C e C e e kt The derivative IS written as a product of a function of t and a function of. Since C is arbitrar, ±e C represents an arbitrar nonzero number. We can replace it with C: Ce kt

14 Continuous Growth and Deca If is a differentiable function of t such that >0 and '=k, for some constant k, then Ce kt C = Initial Value k = Proportionalit Constant k 0 : Growth 0 k 1: Deca

15 Continuous Growth and Deca If: The rate of change in the output (/) is proportional to the output (). Then: k or Ce dt kt Your Choice: Remember how to derive the general solution from the differential equation OR memorize the general solution.

Chapter 5: Integrals

Chapter 5: Integrals Section 5.3 The Fundamental Theorem of Calculus Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus: Sec. 5.3: The Fundamental Theorem of Calculus Fundamental