Final answer:
The fundamental solutions of the given differential equation can include e^2x, √x, cos x, and 2^x, as all these functions possess properties that can satisfy the criteria of being solutions to differential equations depending on their form and context.
Step-by-step explanation:
The question asks to identify the real-valued fundamental solutions of the differential equation represented by x'. To determine fundamental solutions, we can consider the properties of the given functions. For example, the function e^2x is a fundamental solution because it satisfies the property that e^(x+y) = e^x * e^y, which allows us to add the exponents when multiplying.
Another function is √x or the square root of x, which can be expressed as a fractional power, satisfying x^2 = √x when mulitplying it by itself. As for the function cos x, it is also a solution because it derives from the oscillatory behavior of certain systems and is often a solution to second-order linear differential equations. Finally, 2^x is an exponential function and can be a solution depending on the differential equation's structure.