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
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.