1. Determine the eigenvalues and corresponding eigenfunctions of the boundary value problem as following (a) y" + ly = 0 (0 < x < L); y'(0) = 0, y(L) = 0.

Ben can wash a car in 2 hours while his younger brother James can do it in 3 hours. How longwill it take to wash the car if they work together?

5. Use the meyhod of undetermined coefficients to solve the differential equation. a) ¹y" + y + y = x² = x b) y" + 3y = -48t²e-3t

Let x (t) x' (t) x' (t) If x (0) x₁(t) = = = = = sin( x ₂ (t) 2 = sin( x1(t) x2(t) ] Put the eigenvalues in ascending order when you enter x₁(t), x₂(t) below. -17x₁(t) 8 x ₁ (t) 1 -3 [:] sin( t) + sin( t) + t) + cos( be a solution to the system of differential equations: and x'(0) t) + + cos( 16 x ₂ (t) 2 7x₂(t) = -28 16 cos ( t) + t) cos( t) + t) " find x(t).