Verify that the given functions form a basis for the space of solutions of the given differential equation. y"+y=0, f1(x)=cosx, f2(x)=sinx

Question

Verify that the given functions form a basis for the space of solutions of the given differential equation.  y"+y=0, f1(x)=cosx, f2(x)=sinx

anonymous · Answer

first do u know what sin and cos is ?

anonymous · Answer

lol kind of juss started learing it

anonymous · Answer

ok sin is o/h   and cos is a/h

anonymous · Answer

ok thanks

jamesj · Answer

Every second order linear homogeneous equation with linear coefficients
$$ y'' + ay' + by = 0 $$
has a two dimensional solution space.  That is, there exist two linearly independent solutions to the equation.  If we call them y_1 and y_2, then the general solution of the equation is
$$ y = c_1y_1 + c_2y_2 $$

Hence, for your problem, it is sufficient to show that
1. cos x and sin x are solutions of the differential equation; and
2. cos x and sin x are linearly independent

You can show number 1 by just showing cos x and sin x satisfy the differential equation.  Number 2 I leave to you.