Question

Find the particular solution for the differential equation y^n - 4y’ + 4y = 0 subject to the given conditions that y(0) = 12 and y’(0) = -3.

Answer 1

To find the particular solution for the given differential equation, you need to rewrite the equation in terms of y(0) and y'(0) and then solve the characteristic equation. Finally, plug in the values of y(0) and y'(0) into the particular solution formula.

Step-by-step explanation:

1. Rewrite the differential equation in terms of y(0) and y'(0) to obtain the characteristic equation: y^n - 4y' + 4y = (y-4)(y-4y'-1) = 0.
2. Solve the characteristic equation to find the values of y(0) and y'(0) that satisfy the equation. In this case, y(0) = 4 and y'(0) = 1.
3. Plug in the values of y(0) and y'(0) into the particular solution formula: y = y(0) + y'(0) * t = 4 + t.