Final answer:
To verify the given function y = 7e^x - 3e^(-x) is a solution of the differential equation y' = y + 6e^(-x), substitute the function and its derivative into the equation.
Step-by-step explanation:
To verify that a given function is a solution of a differential equation, we need to substitute the function and its derivatives into the differential equation and check if the equation holds true. In this case, we have the differential equation y' = y + 6e^(-x) and the function y = 7e^x - 3e^(-x). To verify, we substitute y and y' into the differential equation:
y' = (7e^x - 3e^(-x))' = 7e^x + 3e^(-x)
y + 6e^(-x) = 7e^x - 3e^(-x) + 6e^(-x) = 7e^x + 3e^(-x)
Since y' = y + 6e^(-x) is true, we can conclude that the given function y = 7e^x - 3e^(-x) is a solution of the given differential equation.