EXAMPLE 7 Flow from a tank Imagine a large cylindrical tank with cross-sectional area A. The bottom of the tank has a circular drain with cross-sectional area a. Assume the tank is initially filled with water to a height (in meters) of h(0) = H (Figure 9.7). According to Torricelli's law, the height of the water t seconds after the drain is opened is described by the differential equation h'(t) = -kVh, where t = 0, k = A and g = 9.8 m/s is the acceleration due to gravity. a. According to the differential equation, is h an increasing or decreasing function of t, for t 2 0? b. Verify by substitution that the solution of the initial value problem is A() = ( Vĩ - )". c. Graph the solution for H = 1.44 m, A = 1 m², and a 0.05 m?. с. d. After how many seconds is the tank in part (c) empty?

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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.