A tank contains 240 liters of fluid in which 50 grams of salt are dissolved. Pure water is then pumped into the tank at a rate of 6L/min; the well mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.
The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial value problem. y = c1e^4x + c2e^-x, (-inf, inf); y'' - 3y' - 4y = 0, y(0) = 1, y'(0)=3
The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial value problem. y=c1 + c2cos(x) + c3sin(x), (-inf, inf); y'''+ y' = 0, y(pi)=0, y' (pi) = 6, y''(pi) = -1
Find the largest interval which includes x = 0 for which the given initial value problem has a unique solution. (Enter your answer using interval notation.) (x-3)y'' + 5y = x, y(0) = 0, y'(0) = 1
Determine whether the given set of functions is linearly independent on the interval (-inf, inf). f1(x) = x, f2(x) = x^2, f3(x) = 3x - 6x^2
Determine whether the given set of functions is linearly independent on the interval (-inf, inf). f1(x) = e^x, f2(x) = e^-x, f3(x) = sinh(x)
Consider the differential equation y''- y' - 12y = 0 Verify that the functions e^-3x and e^4x form a fundamental set of solutions of the differential equation on the interval (-inf, inf). The functions satisfy the differential equation and are linearly independent since the Wronskian W(e^-3x, e^4x)
Consider the differential equation y'' - 2y' + 10y = 0 Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(e^xcos 3x, e^xsin 3x)
The indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5) in section 4.2, instructed to find a second solution y2(x). y'' + 100y = 0; y1 = cos(10x)
The indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5) in section 4.2, As instructed to find a second solution y2(x). 9y'' - 30y' + 25y = 0; y1 = e^5x/3