Question

Find the general solution to the homogeneous differential equation (d²y/dt² - 15dy/dt = 0). The solution can be written in the form (y = c_1eʳ_1t + c_2eʳ_2t) with (r_1 < r_2).

Answer 1

The general solution to the homogeneous differential equation (d²y/dt² - 15dy/dt = 0) is:

y = c₁e^(5t) + c₂e^(3t)

where c₁ and c₂ are arbitrary constants.

Step-by-step explanation:

Characteristic equation:

The characteristic equation is obtained by assuming a solution of the form y = e^(rt). Substituting this into the differential equation, we get:

r²e^(rt) - 15re^(rt) = 0

Dividing both sides by e^(rt), we get:

r² - 15r = 0

Roots of the characteristic equation:

Factoring the equation, we get:

r(r - 15) = 0

Therefore, the roots of the characteristic equation are:

r₁ = 0

r₂ = 15

General solution:

Since we have two distinct real roots, the general solution of the differential equation is:

y = c₁e^(r₁t) + c₂e^(r₂t)

Substituting the values of r₁ and r₂, we get:

y = c₁e^(0t) + c₂e^(15t)

Simplifying the expression, we get:

y = c₁ + c₂e^(15t)

Linear independence:

Since the solutions e^(0t) and e^(15t) are linearly independent, we can add a constant term to the first solution to obtain the general solution:

y = c₁e^(5t) + c₂e^(3t)