6. For the differential equation 2 dy+xEy=x (1):3 dx g I a. Make the usual substitution, x = of to restructure this equation as a differential equation with constlant coefficients with respect to the variable t. Show all work! b. Solve the resultant homogenous part of the differential equation [as a function of ti found in part a. "the easvr wav\" since it is a second order differential equation with constant coefficients. c. Solve for the particular solution using variation of parameters of the differential equation in terms of the variable t as constructed in part a. Note, use the technique as discussed in lectures and homework! d. Write the entire solution vit] to the inhomogeneous differential equation in the variable t, that is vit] found in parts b. and c. and write it as a function of the variable x. that is vix} based on the substitution used in part a.? e. 1I.i'I.."rite the solution in part d. as a function of x for the following initial conditions y'iL'iID. [dyi =]-| d}; \\'C f. Find a series solution for the homogenous equation solved in part b. above. g. Comment on the solutions found in parts f. with respect to the Tavlor series expansions [will need to construct} of the solutions found in part b. h. Solve for the particular solution in part c. above using the Green's function approach.