Joseph M Mahaffy h jmahaffysdsuedu i Lecture Notes Exact and Bernoulli

Joseph m mahaffy h jmahaffysdsuedu i lecture notes

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Joseph M. Mahaffy, h [email protected] i Lecture Notes – Exact and Bernoulli Differential — (18/26) Introduction Exact Differential Equations Bernoulli’s Differential Equation Logistic Growth Equation Alternate Solution Bernoulli’s Equation Bernoulli - Logistic Growth Equation 1 Alternate Solution - Logistic Growth Equation dP dt = rP 1 - P M , P (0) = P 0 This is rewritten dP dt - rP = - r M P 2 Consider a substitution u = P 1 - 2 = P - 1 , so du dt = - P - 2 dP dt Multiply the logistic equation by - P - 2 , so - P - 2 dP dt + rP - 1 = r M or du dt + ru = r M Joseph M. Mahaffy, h [email protected] i Lecture Notes – Exact and Bernoulli Differential Equations — (19/26) Introduction Exact Differential Equations Bernoulli’s Differential Equation Logistic Growth Equation Alternate Solution Bernoulli’s Equation Bernoulli - Logistic Growth Equation 2 Alternate Solution (cont): With the substitution u ( t ) = - 1 P ( t ) , the new DE is du dt + ru = r M , which is a Linear Differential Equation With our linear techniques, the integrating factor is μ ( t ) = e rt , so d dt ( e rt u ( t ) ) = r M e rt so e rt u ( t ) = e rt M + C or u ( t ) = 1 M + Ce - rt or 1 P ( t ) = 1 M + Ce - rt Joseph M. Mahaffy, h [email protected] i Lecture Notes – Exact and Bernoulli Differential — (20/26)

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Introduction Exact Differential Equations Bernoulli’s Differential Equation Logistic Growth Equation Alternate Solution Bernoulli’s Equation Bernoulli - Logistic Growth Equation 3 Alternate Solution (cont): Inverting this gives P ( t ) = M 1 + MCe - rt The initial condition P (0) = P 0 , so P 0 = M 1+ MC or C = M - P 0 P 0 M It follows that P ( t ) = MP 0 P 0 + ( M - P 0 ) e - rt This solution is MUCH easier! Joseph M. Mahaffy, h [email protected] i Lecture Notes – Exact and Bernoulli Differential Equations — (21/26) Introduction Exact Differential Equations Bernoulli’s Differential Equation Logistic Growth Equation Alternate Solution Bernoulli’s Equation Bernoulli’s Equation 1 Definition A differential equation of the form dy dt + q ( t ) y = r ( t ) y n , where n is any real number, is called a Bernoulli’s equation Define u = y 1 - n , so du dt = (1 - n ) y - n dy dt Joseph M. Mahaffy, h [email protected] i Lecture Notes – Exact and Bernoulli Differential — (22/26) Introduction Exact Differential Equations Bernoulli’s Differential Equation Logistic Growth Equation Alternate Solution Bernoulli’s Equation Bernoulli’s Equation 2 The substitution u = y 1 - n suggests multiply by (1 - n ) y - n , changing Bernoulli’s Equation to (1 - n ) y - n dy dt + (1 - n ) q ( t ) y 1 - n = (1 - n ) r ( t ) , which results in the new equation du dt + (1 - n ) q ( t ) u = (1 - n ) r ( t ) This is a 1 st order linear differential equation , which is easy to solve Joseph M. Mahaffy, h [email protected] i Lecture Notes – Exact and Bernoulli Differential Equations — (23/26) Introduction Exact Differential Equations Bernoulli’s Differential Equation Logistic Growth Equation Alternate Solution Bernoulli’s Equation Example: Bernoulli’s Equation 1 Example: Consider the Bernoulli’s equation: 3 t dy dt + 9 y = 2 ty 5 / 3 Solution: Rewrite the equation dy dt + 3 t y = 2 3 y 5 / 3 and use the substitution u = y 1 - 5 / 3 = y - 2 / 3 with du dt = - 2 3 y - 5 / 3 dy dt Multiply equation above by - 2 3
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