Determine whether the given function is a solution to the given differential equation. y=cosx+x8 , d^2y/dx^2 + y = x^8 + 56x^6.

The correct question is:

Determine whether the given function is a solution to the given differential equation. y = cosx + x^8; d²y/dx² + y = x^8 + 56x^6

Step-by-step explanation:

Given the differential equation

d²y/dx² + y = x^8 + 56x^6.

Suppose y = cosx + x^8 is a solution, then differentiating y twice, and adding it to itself, must give the value on the right hand side of the differential equation.

Let us differentiate y twice

y = cosx + x^8

dy/dx = - sinx + 8x^7

d²y/dx² = - cosx + 56x^6

Now,

d²y/dx² + y = - cosx + 56x^6 + cosx + x^8

= 56x^6 + x^8

Therefore,

d²y/dx² + y = x^8 + 56x^6

Which shows that y = cosx + x^8 is a solution to the differential equation.