Verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition I for each solution.

1. y'' + y = 2 cos x  2 sin x; y = x sin x + x cos x

2. y'' + y = sec x; y = x sin x + (cos x)ln(cos x)

3. x²y'' + xy' + y = 0; y = sin(ln x)

4. x²y'' + xy' + y = sec(ln x);

y = cos(ln x) ln(cos(ln x)) + (ln x) sin(ln x)