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Question a]: Define separable differential equation. Solve separable differential equation e11+fll - 21th +e\") = [I b} What do you mean by exact differential equation. Solve exact differential equation by substitution 31 = w x 21+y3 a} What is meant by orthogonal trajectories? Solve orthogonal trajectories cl 2 the family of curves % + = 1 where a is constant and ii. is parameter b]: Why does variation of parameters work?5olve the given differential equation by using method of 1variation of parameters {1 + 4y' + 13y = 3e\" ch:I Sin 3:: a} Discuss simple pendulum in detail. At what point in the motion of a simple pendulum is the string tension greatest and the least ? In each case give the reasoning behind your answer. b} How does a RLE circuit work? How do you find the inductance of an RLE cucu? c] Why we need fourier transformation? Find Fourier series for e" in the interval rr.~= x 1: n, a} Define complex numbers, why we need complex no's? Prove that E1i= E a: 1 J b} Wf'fat is the difference between initial value and boundary value problems' Solve initial value problem y' = xy; yfji = 1 c] Can any function be represented as a power series? Find the power series solutions in Ewer of x of differential oation y"-4y =