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Warm-up It’s as easy as 1-2-3! 1)Start by separating variables. 2)Integrate both sides. 3) Solve for dy/dx. Solve = k(y – 80) This represents Newton’s Law of Cooling, where y is the core temperature of an object and 80 is the ambient temperature in Fahrenheit degrees.

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Homework Review Anticipated Problem (P.368 #74): When an object is removed from a furnace, its core temperature is 1500 deg. F. The ambient temperature is 80 deg. F. In one hour the core cools to 1120 deg. F. What is the core temperature after 5 hours?

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Differential Equations – Part 2 Separation of Variables Text – Section 5.7

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Verifying Solutions In the last class, we learned that the general solution of a differential equation could be made particular by using initial conditions to solve for any unknown constants. You can also verify that a solution, whether general or particular, satisfies a differential equation by plugging the solution into the original equation. Note: When we plug a solution, we must plug for all occurrences of y and its derivatives in the original equation.

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Examples of Verifying Solutions Determine whether the given function is a solution of the differential equation, y” – y = 0. a)y = sin x Because y’ = cos x and y” = -sin x, y” – y = -sin x – sin x = -2sin x ≠ 0 -> NO Solution. b)y = 4e -x Because y’ = -4e -x and y” = 4e -x, y” – y = 4e -x – 4e -x = 0 -> Solution! b)c) y = Ce -x Because y’ = -Ce -x and y” = Ce -x, y” – y = Ce -x – Ce -x = 0 -> Solution (for any value of C)

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Your Turn! Determine whether the given function is a solution of the differential equation, xy’ – 2y = x 3 e x a)y = x 2 b)y = x 2 (2 + e x )

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Solving a Differential Equation and Finding a Particular Solution Solve the equation, xy’ – 3y = 0. Find the particular solution, if y = 2, when x = -3.

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Solving a Differential Equation and Finding a Particular Solution Solve the equation, xy dx + (y 2 – 1) dy = 0 Find the particular solution, if y(0) = 1.

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Finding a Particular Solution Curve Find the equation of the curve that passes through the point (1,3) and has a slope of y/x 2 at the point (x,y).

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Applications #1 - The rate of change of the number of coyotes N(t) in a population is directly proportional to 650 – N(t), where t is the time in years. When t = 2, the population has increased to 500. Find the population when t = 3.

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Applications #2 – Describe the orthogonal trajectories for the family of curves given by y = C/x for C ≠ 0. – > xy = C – > Implicit differentiation: xy’ + y = 0 – > = -, slope of given family – > What is the slope of the orthogonal family?

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