Euler\'s Equation: Solve t2 y\" - 4ty\' + 6y = 0, t > 0, y(l) = 3, y\'(l) = 7 As

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Euler's Equation: Solve t2 y" - 4ty' + 6y = 0, t > 0, y(l) = 3, y'(l) = 7 Assume that y(x) = anxn satisfies the equation y" - 5y' = 0. Find the recurrence relation for the coefficients an, n = 0, 1, 2..... Use the Laplace Transform to solve the given initial value problem : y" + y' - 6y = 0, y(0) = l, y'(0) = 0. Find the general solution of the given system of differential equations: X = X' *Extra** : X = X' Solve the given Boundary Value Problem: y" + 3y = 0 , y(0 ) = 1, y(l) = 0 Find the eigenvalues mu2 and corresponding eigenfunctions of y" + mu2 y = 0 if y(0) = 0 and y(l) = 0 A tank initially contains 500 liters of pure water. A mixture containing concentration of 10 gram/liter of salt enters the tank at the rate of 20 liter/min, and well-stirred mixture leaves the tank at the same rate of 20 liter/min. Find the expression for the amount q(t) of salt at any time t and find the limiting amount of salt as t rightarrow infinity Solve the given differential equation using the method of Integrating Factors and determine how the solution behaves at infinity y'+2y = 4t, y(0) = 3, y' + (2/t) y = 8 t +t-2 t > 0, y(l) = 2 *EXTRA* y'- l/t = t2cos(t - l) + 8t4, y(l) = 5 ** Bernoulli Equation : Solve y' + (1/t) y = -t3 y3, t > 0, y(l) = 3 Solve the given separable differential equation in explicit form, plot the graph of the solution and determine the interval in which the solution is defined: y'(x) = (4 - x )/y, y(4) = 3 Solve the given Exact Differential Equation: (2 xy3 + y)dx + (3x2y2 + x + l)dy = 0 y(2) = 1 Find the Wronskian of: a) f(t) = et , g(t)= t et , b) f(t) = e6t , g(t) = e -3t Solve the given Initial Value Problem and describe the behavior of y(t) as t rightarrow infinity y" + y' = 0, y(0) = 5, y'(0) = - 3, y" - y' - 6y = 0 y(0) = 2, y'(0) = 1, y" + 4y' + 4y = 0, y(0) = 1, y'(0) = - 1 Solve y" + 9y = 0, y(0) = 4, y'(0) = 9 and write the solution in the form y(t) = R cos (omega 0t - delta) if W( f, g ) = 3 e8t and f(t) = e4t, find g(t). Find the general solution of y"+ 2y' + 10y = 0; y(4) + 4y" = 0; y'" - 2y" - y' + 2y = 0 Find the general solution of y" +4y' + 3y = 3t + 7, y" - 4y' + 3y = 6e2t

Explanation / Answer

y'' + 3y = 0 auxiliary eq. is m^2 + 3 = 0 or m = sqrt(3) i, -sqrt(3) i solution is y = ACos(sqrt(3)x) + BSin(sqrt(3)x) put y = 1, x = 0 1 = A put y = 0 x = 1 0 = Cos(sqrt(3)) + BSin(sqrt(3)) or B = -cot(sqrt(3)) y = Cos(sqrt(3)x) - Cot(sqrt(3))Sin(sqrt(3)x)

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