In each of the following questions choose one appropriate answer. No justificati

Question

In each of the following questions choose one appropriate answer. No justification for your answer is necessary. (i) The differential equation y" + 5y' - 9y = e^t cos (y^3) is a (a) linear differential equation (b) non-linear differential equation. (ii) The Laplace transform, L {f(t)} of a function f (t) is given by (a) Integral^+infinity_0 f (e^st) dt (b) Integral^+infinity_0 e^s^2 In (t) f (t) dt (c) Integral^+infinity_0 e^-st f (t) dt (iii) The convolution of two functions f(t) and g(t) is given by (a) Integral^t_0 f (x) g (x) dx (b) Integral^+infinity_0 e^-sf (t) g (t) dt (c) Integral^t_0 f (t - x) g (x) dx (iv) The characteristic polynomial of the matrix A = [2 1 3 7] is given by: (a) set ([2 - r 3 1 7 - r]) (b) det ([2 - r 3 - r 1 - r 7 - r]) (c) det ([2 - r 3 1 - r 7]) Furthermore, given an eigenvalue r_0 of A, a vector xi_0 is called an eigenvector of A corresponding to r_0, if it satisfies the equation (a) A xi_0 = 0 (b) (A - r_0I) xi_0 = 0 (c) (r_0 A - I) xi_0 = 0. (v) The Wronskian W (x(t), y(t)) of a pair of two-dimensional vector functions x (t) = [x_1 (t) x_2 (t)], y (t) = [y_1 (t) y_2 (t)] is given by (a) x_1 (t) y_2 (t) - x_2 (t) y_1 (t) (b) x_1 (t) y'_2 (t) - x_2 (t) y'_1 (t) (c) x'_1 (t) y'_2 (t) - x'_2 (t) y'_1 (t)

Explanation / Answer

I. Non-Linear- Because it contain y^3 term in it

II.(C). -it is the general form of Laplace transform

III.(C). -convolution theorem has been applied to Laplace and this is the result

IV. (A),(B) Characteristic Polynomial Equation changes the term 11,22,33...etc by subtracting from Ro.

V.(B). Differential Equation formula

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