Suppose U solves the heat equation on the real lineUt = 4Uxx, x ∈ Rwith initial valueU(x, 0) = (4, x ≤ 02, x > 0.(i) Use the Fourier-Poisson formula to give an explicit expression for the solutionU.(ii) Describe the qualitative behaviour of U in this case as t → ∞ and plot outthe solution at several instants of time to explain your answer. What is the limitof U as t → ∞?

A model for the velocity v al time t of a certairs object failing under the influence of gravity in a viscous medium is given by the equation -3-- From the diroction lield shown in the figure to the right, skelch the dv solutions with the initial conditions v(0) = 10, 12 , and 19. Wny is the value v 12 called the 'terminal velocity"? Choose the coIreci sketch of the solutions with the initial conditions v(0) = 10, 12, and 19. O A OB. Oc.

solve asapSolve using Methods of solution to High Order Differential EquationOr using Laplace Transform

What is dufferential equation of x'=4x-y and y'=-4x+4y when t=0, x(0)=1 and y(0)=1