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Section 6.2 – Differential Equations (Growth and Decay)

Section 6.2 – Differential Equations (Growth and Decay). Reminder: Directly Proportional.

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Section 6.2 – Differential Equations (Growth and Decay)

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  1. Section 6.2 – Differential Equations (Growth and Decay)

  2. 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, yis proportional to x if k is a constant and:

  3. 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 yis inversely proportional to x and if k is a constant:

  4. 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:

  5. 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: The first derivative is the product of… and a function in terms of y. a function in terms of x…

  6. Examples Which first order differential equations below are separable? Separable because it is a product of a function of x (sinx) and a function of y (y2) 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 ofy

  7. Separation of Variables If a first order differential equation is separable, use the following solution method: • Make sure the differential equation is written as a product of a function of x and a function of y. • Move all of the y terms on one side and all of the xterms on the other; this includes the dx and dy. • Integrate both sides. • Solve for y(if possible).

  8. Example 1 Find the general solution to: Now use separation of variables to find the general solution. Is this a separable equation? YES. The derivative can be written as a product of a function of x and a function of y. A C on both sides would be redundant. C is arbitrary, so there is no difference between 2C and C.

  9. Example 2 In this example, t = x. Solve the initial value problem: Now use separation of variables to find the general solution. The derivative IS written as a product of a function of t and a function of y. Since C is arbitrary, ±eCrepresents an arbitrary nonzero number. We can replace it with C: Now use the initial condition to find the particular solution:

  10. Example 3 Solve the initial value problem: 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. Now use the initial condition to find the particular solution: Not every equation can be solved for y.

  11. Annual Growth Jason 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? What about if it is compounded continuously? The rate of change in the new output is directly proportional to the previous output.

  12. “Continuous” Growth and Decay If 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: We can use separation of variables to find the general solution for a growth or decay situation.

  13. “Continuous” Growth and Decay Find the general solution to: Now use separation of variables to find the general solution. The derivative IS written as a product of a function of t and a function of y. Since C is arbitrary, ±eCrepresents an arbitrary nonzero number. We can replace it with C:

  14. “Continuous” Growth and Decay If yis a differentiable function of tsuch that y>0 and y'=ky, for some constant k, then C= Initial Value k= Proportionality Constant

  15. “Continuous” Growth and Decay If: Then: The rate of change in the output (dy/dx) is proportional to the output (y). Your Choice: Remember how to derive the general solution from the differential equation OR memorize the general solution.

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