If u(x, t) is the solution of then ди = J²u It მე-2 ა ,0 < x < 1,t > 0, u(x, 0) = 1+x+ sin (πx) cos (πx), u(0, t) = 1, u(1, t) = 2, (a) u() = (b) (ਨੂੰ, ਨੂੰ) = (c) u()+½½-3² = (d) u(1, 1) = e−4 -42

6) If y=x² +1+ f dt , prove that y"=2 - x?

Consider the equation ut = Uxx, 0 0. Suppose u(0, t) = 0, u(1, t) = 0. Suppose u(x, 0) = -3 sin(x) + 2 sin(2x) + 3 sin(3πx) + 4 sin(4x) Fill in the constants in the solution: u(x, t) = help (numbers) е -² sin(x) + e-(2) sin(2x) + -(3π)² sin(3πx) + e e (4) ² sin(4x)

Here i is the imaginary unit.a) Use the Euler formula, e^(iθ) = cos(θ) + i sin(θ), to prove that for any n, we have(cos(θ) + i sin(θ))^n = cos(nθ) + i sin(nθ).b) Suppose y(t) = e^(−2t)e^(it) solves y′′ + By′ + C = 0. What are B and C?

Verify that y(t)=C_(1)sin5t+C_(2)cos5t is a solution of y''(t)+25y(t)=0