Solving by systematic elimination

Question

Assignment:

Q1. Solve the differential equation y⁴ - 2y'' + y'' = 1+e^x using the undetermined coefficients method.

Q2. Solve the differential equation y'' - 2y' + y = e^x / 1+ x² using variation of parameters method.

Q3. Solve the initial value problem ( -∞ , 0): 4x2y'' + y = 0 and with  y (-1) = 2 and y'(-1) = 4.

Q4. Solve by systematic elimination. Here D is d/ dt.

(D + 1)x + (D - 1)y = 2

3x + (D + 2)y = -1

Provide complete and step by step solution for the question and show calculations and use formulas.