Consider the differential equation y'' − y' − 20y = 0. Verify that the functions e−4x and e5x form a fundamental set of solutions of the differential equation on the interval (−[infinity], [infinity]).The functions satisfy the differential equation and are linearly independent since the Wronskian W e−4x, e5x = ≠ 0 for −[infinity] < x < [infinity].Form the general solution. y =

Therefore and are fundamental solution of the given differential equation.

Therefore   and are linearly independent, since

The general solution of the differential equation is

Step-by-step explanation:

Given differential equation is

y''-y'-20y =0

Here P(x)= -1, Q(x)= -20 and R(x)=0

Let trial solution be

Then,   and  

Therefore the complementary function is =

Therefore and are fundamental solution of the given differential equation.

If and are the fundamental solution of differential equation, then

Then   and are linearly independent.

Therefore   and are linearly independent, since

Let the the particular solution of the differential equation is

and

Here , , ,and  

       =0

and

       =0

The the P.I = 0

The general solution of the differential equation is

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