The differential equation below models the temperature of a 94°C cup of coffee in a 24°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 74°C. Solve the differential equation to find an expression for the temperature of the coffee at time t. (Let y be the temperature of the cup of coffee in °C, and let t be the time in minutes, with t = 0 corresponding to the time when the temperature was 94°C.) 1 dy (y- 24) dt 50 Step 1 dy We have dt (y-24), 50 Which separates to become dy = -dt Step 2 Ignoring the constant of integration, integrating the left side gives (remember to use In Ju| where appropriat S dy = y-24 Step 3 Ignoring the constant of integration, integrating the right side gives Sat = dt3D 50 Step 4 We now have Inly - 24| = -t + C.and so ly – 24| = 50 Step 5 Consequently, we write y-K +24, where K- te Step 6 Finally, using y (0) -94 allows us to solve for K- Step 7 The temperature of the coffee at the time is described by the following equation.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.