Question

Consider the following differential equation to be solved by variation of parameters: 4y" - y = ex/2. Find the complementary function of the differential equation.

Answer 1

The complementary function of the differential equation
`4y` is
`y_c = C_1e^((1/2)x) + C_2e^(-(1/2)x)`, where
`C_1` and
`C_2` are constants determined by initial conditions.

Step-by-step explanation:

To find the complementary function of the given differential equation, we need to solve the homogeneous version of the equation, which is
`4y`. This is a second-order linear differential equation with constant coefficients.

We look for solutions in the form
`y = e^(rt)`, which, after substitution into the homogeneous equation, gives us the characteristic equation:

`4r^2 - 1 = 0`

Solving for r, we obtain:

r = ±½

Therefore, the general solution to the homogeneous equation, or the complementary function, is:

`y_c = C_1e^((1/2)x) + C_2e^(-(1/2)x)`

where
`C_1` and
`C_2` are arbitrary constants determined by initial conditions.