Question

find the particular solution y = f(t) for each differential equation.1. dy/dt = 8y and y = -2 when t = 02. dy/dt = -4y and y = 10 when t = 03. dy/dt = 16y and y = 5 when t = 04. dy/dt = -7y and y = -4 when t = 0

Answer 1

For each of these differential equations, we can use the general solution for a first-order linear differential equation of the form dy/dt = ky, which is y(t) = Ce^(kt), where C is a constant.

1. dy/dt = 8y and y = -2 when t = 0:
y(t) = Ce^(8t)
-2 = Ce^(8*0)
-2 = C
y(t) = -2e^(8t)

2. dy/dt = -4y and y = 10 when t = 0:
y(t) = Ce^(-4t)
10 = Ce^(-4*0)
10 = C
y(t) = 10e^(-4t)

3. dy/dt = 16y and y = 5 when t = 0:
y(t) = Ce^(16t)
5 = Ce^(16*0)
5 = C
y(t) = 5e^(16t)

4. dy/dt = -7y and y = -4 when t = 0:
y(t) = Ce^(-7t)
-4 = Ce^(-7*0)
-4 = C
y(t) = -4e^(-7t)