Final answer:
To solve the given differential equation y′′′−9y′′+20y′=48eˣ, we can use the method of undetermined coefficients. Assuming y = Aeˣ, we find that A = 4, so the solution to the equation is y = 4eˣ.
Step-by-step explanation:
To find y as a function of x, we need to solve the given differential equation y′′′−9y′′+20y′=48eˣ for y. We can use the method of undetermined coefficients to solve this equation. Let's assume that y = Aeˣ, where A is a constant. Now, we'll take the derivatives of y and substitute them into the differential equation to find the value of A.
Substituting y = Aeˣ, y' = Aeˣ, y'' = Aeˣ, and y''' = Aeˣ into the differential equation, we get:
Aeˣ - 9Aeˣ + 20Aeˣ = 48eˣ
Combine like terms:
12Aeˣ = 48eˣ
Cancelling the exponential terms, we get 12A = 48. Solving for A, we find A = 4.
Therefore, y = 4eˣ is the solution to the differential equation.