For the following differential equations determine the order of the given differential equation and state whether the equation is linear or nonlinear. (a) t^2 d^2 y/dt^2 + t dy/dt + 2y = sin t: y(0) = 1, y'(0) = 8 (b) (1 + y^2) y"(t) + ty'(t) + y = e^t (c) d^4 y/dt^4 + d^3 y/dt^3 + d^2 y/dt^2 + dy/dt + y = 1 (d) dy/dt + ty^2 = 0: y(1) = 3 (e) y" + sin(t + y) = sin(t) (f) d^3 y/dt^3 + t dy/dt + (cos^2 t)y = t^3 (g) Which of the differential equations above are IVPs (initial value problem)? In each of following, verify that each given function is a solution of the differential equation. Also, are the solutions particular solutions or general solutions? (a) y" + 2y' - 3y = 0: y_1 (t) = c_1 e^-3t, y_2 (t) = c_2 e^t (b) ty' - y = t^2: y = 3t + t^2 (c) d^4 y/dt^4 + 4 d^3 y/dt^3 + 3y = t: y_1(t) = t/3, y_2 (t) = e^-t + t/3
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