Mr. Kuntz' Class
6.1.2 Solving Differential EquationsHW: p.312(25-35), p.322(42,43)p.348(22,29,30)Objective – Solve differential and separable differential equations with initial conditionsWarm UpEvaluate the following integralsx-4x+13xdxe2t-52tdt11+x2dxExtra Integral Rulesekxdx= sinkxdx=coskxdx=Why?Water is leaking out of a right circular cylinder with a radius of 10 cm and a height of 60cm at a rate given by Lt=2+2cost in cm3/min. A hose is filling up the cylinder simultaneously at a rate given by Ht=1+3e-t. There is initially 10 cm of water in the cylinder.How fast is the height of the cylinder changing initially (t = 0)?How much water is in the tank at t=π minutes?On the interval [0,5], when is the amount of water in the tank the greatest? (Use calculator).Differential Equations – General SolutionsSeparate and Integrate!dydx=2x-cosx+2xdydx=x(x2+2)3dydx=y sec2xdydx=2x-x2y2Particular Solutions (INITIAL VALUE PROBLEMS) – Use initial value to find cdydx=x2y3 , y1=-3dydx=y(x-1)2 , y2=1Practice ProblemsSolve each of the following differential equations.dydx=2xy , y1=4dydx=x4y , y1=-1dydx=x24-y , y2=-5Writing Differential EquationsThe rate of change of y with respect to x is directly proportional to y. The initial value of y is 50 and when x = 2, y = 100. Write an equation that relates y in terms of x.The rate at which water is cooling is directly proportional to the temperature of water. When t= 0, the temperature is 70 degrees F, and after 2 minutes the temperature is 56 degrees F. Write an equation for the temperature of the water. ................
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