One solution of the differential equation y y 0 is y cos x

One solution of the differential equation y\"+ y = 0 is y- cos x. A second linearly independent solution is

Solution

Second solution is y=sin(x) We can verify this is a solution first by susbtitutiting

(sin x)\'\'=-sin(x)

Hence it is a solution

Now we check linear independence

a sin(x)+b cos(x)=0

If they are not linearly independent then a or b is non zero let a be non zero

So ,

tan(x)=-b/a but this fixes value of x and for all x we cannot satisfy this hence

a=b=0

And hence two solutions are linearly independent