Question

Apply your knowledge about the shapes of the solutions to the OHO problem to cetermine that tveo calec latiomnlizableantha een fisaorognto frnons (oar v up to 9). (No calculation needed).

Answer 1

Given that Even functions are orthogonal to odd functions Let f (x) rightarrow Even number g (x) rightarrow odd number And h (x) = f (x) middot g (x) Therefore h (x) = will be odd Therefore I = integral^a_-a h (x) dx = integral^a_-a h (x) dx + integral^a_0 h (x) dx To make change t = -x Equation will become integral^a_-a h (-x) dx = -integral^a_-a h (t) dt Therefore I = 0 By this we can conclude that Integrable of even function is orthogonal to any integrable ODE function. By Inner product space to prove Sin nx cos nx are orthogonal It is enough to show < cos nx, sin nx > = 0 i.e. on [-pi, pi] i.e., integral^pi_-pi cos bx sin nx dx = 0 implies integral^pi_-pi cos nx sin nx dx = 1/2 integral^pi_-pi [sin (mx + nx) - sin (mx - nx)] dx = 1/2 [integral^pi_-pi sin (m + n) x dx - integral^pi_-pi sin