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Differential Equations and Seperation of Variables

Differential Equations and Seperation of Variables. dP dt. = .05P.

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Differential Equations and Seperation of Variables

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  1. Differential Equations and Seperation of Variables

  2. dP dt = .05P The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential equation that expresses this fact. Solve it to find an equation that expresses population as a function of time. Rate of change of the population Let P = population t years after the present. *label is people/yr

  3. dP dt = .05P 4

  4. The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential equation that expresses this fact. Solve it to find an equation that expresses population as a function of time. .To check accuracy of equation, find the population after one year given …..” The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year” ; Use the equation we found to check if when t=1 the population comes out right! *when t = 1, P should equal 1050. (1000*.05= 50 plus original 1000) P = 1000 e0.05 t found equation *with this equation, however P = 1051.27 when t = 1. P should equal 1050. 4 Must change up a little!!!! start over

  5. The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential equation that expresses this fact. Solve it to find an equation that expresses population as a function of time. = kP dP dt Constant of proportionality We must find a new constant (k) since 0.05 didn’t work!!! We must find a new constant (k) since 0.05 didn’t work!!! *When t = 0, P = 1000 *when t = 1, P should equal 1050. (1000*.05= 50 plus original 1000)

  6. *When t = 0, P = 1000 = kP = kP dP = k dt P dP dt dP dt *when t = 1, P should equal 1050. *Separate variables in solve ln | P | = k t + C

  7. Solve for k (what we wanted) using above equation and *when t = 1, P = 1050. Now have: |P| = Ce k t *When t = 0, P = 1000 Solve for C using: *You can solve for C as soon as you find antiderivative if there is not an absolute value involved……otherwise solve with C intact AND substitute  j = ± C allowing a drop of absolute value!!!  P = j e k t

  8. therefore: P = 1000e ln t 21 20 In general: y = yo e k t where yo is original value at time, feet, or whatever = 0 Law of exponential change!!

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