Given the differential equation with initial condition shown

Given the differential equation with initial condition shown below: dy/dx = 8 - 4y y(0) = 1 Determine the particular solution.

dy/dx + 4y = 8

writing the homogeneous characteristic equation we get

p + 4 = 0

p = -4

yh(x) = c1*e^{-4x}

Now solving for yp general solution yp = Ax + B

yp\' + 4yp = 8

A + 4Ax + 4B = 8

comparing the coefficients we get

A = 0, B = 4

hence yp = 4

So the final solution is

y = yh + yp = c1*e^{-4x} + 4

y(0) = 1 = c1 + 4

c1 = -3

y = -3e^{-4x} + 4