Find the general solution of the given differential equation. y' + 4x3y = x3 y(x) = ?   Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution.

Q2*) Consider the extremisation of the integral I[y] = √²² F(x,y,y', y") dx x1 when y and y' are prescribed only at x = x1. Derive the so-called 'natural boundary conditions' that must be satisfied at x = x2. Taking a specific example: The functional I [y] is defined by I[y] = √² ((y″)² + y) dx with y(0) = 0 and y'(0) = 0. Write down the fourth-order Euler-Lagrange equation for this problem, stating the four boundary conditions. Find the general solution of the Euler-Lagrange equation, and then impose the boundary conditions to find the extremal.

3 00 By changing to circular coordinates, evaluate foo √²²+v³ dx dy.