So I'm supposed to verify the solution of the differential equation. The solution is y=e^(-x). The differential equation is 3y' + 4y = e^(-x). What is the problem asking me to do here?
plug in y to see whether it satisfies the differential equation
y = e^(-x)
y' = -e^(-x)
3y' + 4y = -3e^(-x) + 4e^(-x) = e^-x
so, it does satisfy it.
Oh, okay. I get it now. Thanks!
The problem is asking you to verify if the given solution, y = e^(-x), satisfies the given differential equation, 3y' + 4y = e^(-x). To do this, you need to substitute the solution y = e^(-x) into the differential equation and check if the equation holds true. Let's go through the steps to verify the solution.
Step 1: Find the derivative of y with respect to x, denoted as y'.
The derivative of y = e^(-x) can be found using the chain rule. Since e^(-x) is an exponential function, its derivative is also e^(-x).
So, y' = d(e^(-x))/dx = -e^(-x).
Step 2: Substitute the solution y = e^(-x) and its derivative y' = -e^(-x) into the differential equation 3y' + 4y = e^(-x).
Now, we can substitute y = e^(-x) and y' = -e^(-x) into the differential equation:
3(-e^(-x)) + 4(e^(-x)) = e^(-x).
Simplifying this equation further:
-3e^(-x) + 4e^(-x) = e^(-x).
Combine like terms:
e^(-x) = e^(-x).
Step 3: Analyzing the equation.
After simplifying, we can see that both sides of the equation are equal. This means that when we substitute y = e^(-x) and y' = -e^(-x) into the differential equation, it satisfies the equation. Therefore, the solution y = e^(-x) is verified to be correct for the given differential equation 3y' + 4y = e^(-x).