Consider the heat equation Ut = AUxx, a €R>⁰, 00 with boundary conditions ux (0, t) = ux (L, t) = 0, t> 0.

Suppose that a projectile is fired straight upward from the surface of the earth with initial velocity vo < 2GM/R. Then its height y(t) above the surface satisfies the initial value problem d²y GM y(0) = 0, y'(0) = vo. dr2 y + R)2 Substitute dv/dt = v(dv/dy) and then integrate to obtain v² = v3 - 2GMY R(R+ y) for the velocity v of the projectile at height y. What maxi- mum altitude does it reach if its initial velocity is 1 km/s?

The region bounded by f(x) 2x² + 12x + 14, x = = and y 0 is rotated about the y-axis. Find the volume of the solid of revolution. = Find the exact value; write answer without decimals. 0,

Solve the heat equation Ut = CUxx1 on the domaion 0 ≤ x ≤ 3, t > 0, u(x, 0) = ² - ³x², ux(0, t) = 0, X 3 2' subject to ux (3, t) = 0.