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Extra Problems (Differential Equations)
11 A population has a net birthrate of 2% per year and a constant net immigration rate of 50,000 per year. At time t = 0, the population is one million. Find the population y as a function of t. 12 Repeat Problem 11 for a net immigration rate of 50,000 per year (that is, emigration exceeds immigration by 50,000 per year). 13 Show that the initial value problem y' = cos (y^{2} + t), y(0) = 1, has a unique solution for 0 ≤ t < ∞. 14 Show that the initial value problem y' = 1/(2 + sin y), y(0) = 1, has a unique solution for 0 ≤ t < ∞. 15 Find the general solution of y"  5y' + 4y = 0. 16 Find the general solution of y" + 400y = 0. 17 Find the general solution of y"  4y' + Sy = 0. 18 Find the general solution of y"  14y' + 49y = 0. 19 Solve y" + 4y'  5y = 0, y(G) = 0, y'(0) = 1. 20 Solve y"  20y' + 100y = 0, j(0) = 1, y'(0) = 0. 21 A massspring system mx" + bx' + kx = 0 has mass m = 2 gm and constants b = 6 and k = 5. At time t = 0, its position is x(0) = 10 and its velocity is x'(0) = 0. Find its position x as a function of t. 22 Work Problem 21 if the system is subjected to constant external force of 3 dynes. 23 Find the general solution of y"  5y' + 4y = 2 + t. 24 Find the general solution of y" + 400y' = e^{t}. 25 Find the general solution of y"  4y' + 8y = cos t. 26 Find the general solution of y"  14y' + 49y = t^{2}. 27 Solve y" + 4y'  5y = 26 sin t, y(0) = 0, y'(0) = 0. 28 Solve y"  20y' + 100 = e^{10t}, y(0) = 0, y'(0) = 0.


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