Problem #1: In a tubular reactor, diffusion of reactant 'A' is Reaction accompanied by homogeneous second-order reaction (A=>B) at a rate = kC2. Here, k is the 2nd-order reaction rate constant and C is the concentration. Assume that C is only a function of the distance along the tube axis (x). Co The diffusion coefficient of 'A' is not a constant IN OUT and depends on the concentration of 'A' per the expression: D=aC", where a is a constant. Your goal is to analyze the diffusion-reaction x=0 x x+Ax x- 00 system by completing the following tasks: (a) First, setup a mass balance and derive a differential equation that represents the concentration distribution of the reactant 'A' in the tubular reactor. (b) Using the method of substitution, and assuming P=dC/dx, simplify the 2nd order differential equation derived in (a) to a form that represents the Bernoulli's 1st order differential equation (refer to class notes #6 on Canvas). Identify the value of n in Bernoulli's equation. (c) Using the method of substitution, and assuming V=P1-", reduce the equation derived in (b) to a 1st order ordinary linear differential equation. Solve this equation using the integration factor method (class notes #3 on Canvas). Derive an expression for P (=dC/dx) in terms of constants k and a and the concentration C, and show that this expression has the general form: P = acB + (CI) In the above equation, (CI) is the first constant of integration and a, B, y are constants. Please provide expression for a, B, y based on derivation in (c) above. Problem #2: Solve the following differential equation and report the complete solution y = f(x): y" + 4y = e2ix (where i = V-1)