Find a homogeneous linear differential equation with constant coefficients whose general solution is given. y = c1 cos (x) + c2 sin (x) + c3 cos (6x) + c4 sin (6x)

The homogeneous differential equation is

d^4y/dx^4 + 37d²y/dx² + 36 = 0

Or

y^ (iv) + 37y'' + 36 = 0

Step-by-step explanation:

We want to find a homogeneous linear differential equation with constant coefficients whose general solution is given as

y = c1 cos (x) + c2 sin (x) + c3 cos (6x) + c4 sin (6x)

This, we are working in reverse. Instead of having a differential equation, and writing and solving it's characteristic equation, before writing it's general solution, we are already given the general solution, so, we work in reverse.

Given

y = c1 cos (x) + c2 sin (x) + c3 cos (6x) + c4 sin (6x)

By observation, we have

(1) Homogeneous differential equation of the fourth order, because of the number of constants

(2) Two pairs of complex roots, that's why we have cosines, and sines.

Knowing that if

m = a ± ib

The general solution is

y = (e^ax) (c1 cos bx + c2 sin bx)

We can now write for

y = [c1 cos (x) + c2 sin (x) ] + [c3 cos (6x) + c4 sin (6x) ]

As

m = 0 ± i

and

m = 0 ± 6i

So that we have

m = √ (-1)

m = √ (-36)

m² = - 1

m² = - 36

m² + 1 = 0

m² + 36 = 0

The auxiliary equation is therefore

(m² + 1) (m² + 36) = 0

m^4 + 36m² + m² + 36 = 0

m^4 + 37m² + 36 = 0

Finally, the homogeneous differential equation is

d^4y/dx^4 + 37d²y/dx² + 36 = 0

Or

y^ (iv) + 37y'' + 36 = 0