5. Water is being pumped continuously into a tank at a rate...

5. Water is being pumped continuously into a tank at a rate that is inversely proportional to the amount of water in the tank; that is: dy/dt = k/y , where y is the number of gallons of water in the tank after t minutes (t ? 0). Initially there were 5 gallons of water in the tank and after 3 minutes there were 7 gallons. a. What is the value of k in the equation dy/dt = k/y? b. How many gallons of water were in the tank at time 18 minutes? 6. Consider the differential equation dy/dx = x^4(y - 2) , a. On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. b. Describe all points in the xy-plane for which the slopes are negative c. Find the particular solution y=f(x) to the given differential equation with an initial condition f(0)=0

Transcript

00:01 Hi, from the question given that, suppose water is being pumped continuously into a tank at a rate that is inversely proportional to the amount of water in the tank, that is, d .y by dt is equal to k by y, where y is the number of gallon of water in the tank after one minute, initially there were five gallons of water in the tank and after three minutes.

00:34 So if t is equal to zero then the water level will be five and if t is equal to three minutes then the water level will be seven seven gallons of water so here we need to find the value of k so this is the value of y so first separate the variable by using the variable separate method so y divide is equal to k times d t now integrate on both side so we obtain y squared divided by two is equal to k t plus the integrating constant c so y is equal to under root of 2 k t plus c1 now substitute the initial condition so y of 0 is equal to under root of 2 k times of 0 plus c1 which is equal to 5 so c1 is equal to 25 similarly substitute y of 3 is equal to 7 so y of 3 is equal to under root of 2 k times of 3 plus 25 which is equal to 7 now simplify it further so we obtained 6 k is equal to 24 so k is equal to 4 hence the value of k is 4 and also we need to find the water level at t is equal to 18 minutes so y of 18 is equal to under root of 2 times of k is 4 times of 18 plus 25 so simplify this we have 10 under root of 116 which is equal to 13 hence we conclude that the value of k is equal to 4 and the water level at 18 minutes that is y of 18 is equal to 13 gallons now we move on to part b so for part b it is given that dy by d x is equal to x to the power of 4 times of y minus 2 so under this we need to create the slow field of the given differential equation so the graphical representation is as follows so this is the graphical representation of the given slope field.

03:22 Now we move on to the next one.

03:25 So here, from this clearly, that is x, y and y dash, the points are 1, 1, 1, 0, 0, 0, 0, negative 1, negative 1, negative 1, negative 1, negative 1, negative 1, negative 1, negative 1, negative 1, negative 1, negative 1, negative 1, 1.

03:48 Similarly the value of y is 1 2 3 0 1 2 3 0 1 2 3 0 1 2 0 0 0 0 0 0 0 0 0 negative 2 2...

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