Question

Observe that equation (3) has constant coefficients. If y1(x) and y2(x) form a fun- damental set of solutions of equation (3), show that y1(ln t) and y2(ln t) form a fundamental set of solutions of equation (2). Show that they are both solutions and also form a fundamental set.

Answer 1

The equations (2) and (3) you referred to are unavailable, but it is clear that you are trying to show that two set of solutions y1 and y2, to a (second-order) differential equation are solutions, and form a fundamental set. This will be explained.

Answer:

SOLUTION OF A DIFFERENTIAL EQUATION.

Two functions y1 and y2 are set to be solutions to a differential equation if they both satisfy the said differential equation.

Suppose we have a differential equation

y'' + py' + qy = r

If y1 satisfies this differential equation, then

y1'' + py1' + qy1 = r

FUNDAMENTAL SET OF DIFFERENTIAL EQUATION.

Two functions y1 and y2 are said to form a fundamental set of solutions to a second-order differential equation if they are linearly independent. The functions are linearly independent if their Wronskian is different from zero.

If W(y1, y2) ≠ 0

Then solutions y1 and y2 form a fundamental set of the given differential equation.