Verify that the given functions form a basis for the spacesolutions of the given

Question

Verify that the given functions form a basis for the spacesolutions of the given differential equation.

y'-2xy = 0 ,

Example answer format for similarproblem

y''+y = 0,,

We easily verify that , are solutions and hencebelong to the kernel of the operator

Furthermore, we see that these functions are linearlyindependent since their Wronskian

does not vanish for all x. Hence the general solution of theequation is

***I am stuck because there is only one solution f(x) given not2 like the example, I don't see how I can do the similarthing***

Thanks a lot for your help

Explanation / Answer

Since the differential equation is of the first order, thereis only one distinct function needed to form the basis of thesolution space, it is sufficient to show that the given solution isindeed the correct solution of the stated problem. Best Regards, CR

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