Question

Consider the following ODE: y ′ +3y ′−10y=12 ᵉ²ᵗFind the general solution. Show your work: Show all work needed to find a particular solution to the equation, as well as the general solution.

Answer 1

To solve this ODE, we use the method of integrating factors. The general solution is y = -12e^(-2t) + Ce^(3t).

Step-by-step explanation:

To solve the given ordinary differential equation (ODE), we can use the method of integrating factors. First, we rearrange the equation to the standard form:

y' + 3y - 10y = 12e^(2t)

The integrating factor is found by taking the exponential of the integral of the coefficient of y, which in this case is -3. So the integrating factor is e^(-3t). We multiply both sides of the equation by the integrating factor to get:

(e^(-3t)y)' = 12e^(-t)

Integrating both sides gives us:

e^(-3t)y = -12e^(-t) + C

Solving for y, we obtain the general solution:

y = -12e^(-2t) + Ce^(3t)