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In Exercise 9.2.28 we discussed a differential equation that models the temperature of a $ 95^oC $ cup of coffee in a $ 20^oC $ room. Solve the differential equation to find an expression for the temperature of the coffee at time $ t. $

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Calculus 2 / BC

Chapter 9

Differential Equations

Section 3

Separable Equations

Missouri State University

Campbell University

Idaho State University

Boston College

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

33:32

04:18

In Exercise 28 in Section …

09:19

03:45

06:02

10:56

In Exercise 14 in Section …

solution off problem 38 from exercise 28 in section 9.2. We know that the differential equation the capital t by these multi is equal minus four point or two multiplied by capital T minus 20 here capital t his temperature at a time. Small tea. The capital t by this multi is equal minus four point or two multiplied by capital T minus 20. So decommitted lt Divided boy T minus 20 is equal minus opened or two ds multi. Integrate both sites to get Lynn T Chemical. T minus 20 is equal minus or going toe toe. Small t plus Len Si frieze is a power. Does it these e on both sides. We ought to get e to the power. Lynn Capital T minus 20 is equal to hey to the power minus four point or two. Small tea plus linsi So e to the power. Lynn Chemical T minus 20 is equal toe e to the power Lindsay multiplied boy e to the power minus a point or two. T Using is a property. He it was a bower. Lynn A is equal. A so de minus 20 is equal toe see multiplied by E to the power miners, or point both multi So capital T is equal to 20 plus c multiplied boy e to the power miners open toes multi From the initial condition. Initial temperature is equal to 95 Phylicia's from the question. Therefore, Carrie Tell T is equal to 95 when small T is equal to zero. So, Mr to the initial conditions to get 95 is equal to 20 plus C multiplied by E to the power zero. So 95 minus 20 is equal to see multiplied by one so C is equal to 9. 75. So institutes a value off see to get difficulty is equal toe 20 plus 75 multiplied by E to the power minus 4.2 multi. So there is a will be capital T is equal to 20 plus 75 multiplied by e those of our negative over in tow small

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