Consider the DE y"(x) + p(x)y'(x) + q(x) y(x) = 0. %3D The corresponding "normal form" DE is 1 1 u"(x) + Q(x)u(x) = 0, where Q(x) = q(x) – p(x)' P(x)? %3D %3D (a) The Bessel DE of order 1 is x y" + xy'+ (x-1) y = 0. %3D Find its normal form DE. (b) Explain why the distance between successive zeros of any nontrivial solution of this DE must be greater than T. (Refer to one of the Sturm theorems.)

ee) Using the Laplace transforms to solve the following equations: y" + 3y' + 2y = e*, y (0) = 0, y' (0) = 1 sin 2t, y (0) = y (0) y" + 3y' + 2y = f (t), y (0) = 0, y' (0) = 0 ) y" + y' + 2y = cos (3t), y (0) = y' (0) = 0 ) y" + 2y' + 2y = 2step (t – 3), y (0) = 0, y' (0) = 0 ) y" + 2y' + 10y = 8 (t – 2), y (0) = 1, y' (0) |3D y" + 5y = = 0 -3

(b) Find the derivative of f(x) = g'(x) = h' (x) = f'(x) = ab ab ab a lo b b almo = (x² + 2) (8æ — 5) by using the product rule. Let g(x) = ² + 2 and h (x) = 8x − 5. va a ㅠ sin (a) √a |a| ㅠ √a a ㅠ sin (a) sin (a) [11 183

L(y) = any(n)(x) + An−1 y(n − 1)(x) + where ao, a1, ..., L(y) = 3e²x cos x + 6xe²x (*) Suppose that it is known that L[y₁(x)] = 8xe2x L[y2(x)] = 4e²x sin x L[y3(x)] = 5e²x COS X Find a particular solution to (*). ‚ an are fixed constants. Consider the nth order linear differential equation + a₁ y'(x) + ao y(x) = when y₁(x) 24.xe2x 8e²x cos x when y₂(x) when y3(x) = 40e2x cos x + 120e²x sin x =