This is only one question related to Differential Equations MAT244

3. (25 points). Let 11(23, t) = a(t)b(m) where (1,?) are smooth 11011 identically zero functions where f. 2 U and I E R are independent variables. Also let a be positive and decreasing. Let u solve the heat equation ut=u $3.? a (10) Show that if cuff.) = U for some t, then the nth derivative is alnl) = 0 for all n, that is by smoothness of a we have that a must be identically zero. Assume that b(:r) aE 0 on some open interval (ILL) o (10) Show that the functions (1,6 satisfy the system of ODEs a' = ca b\" = cb on (0, L) for some constant c and explain why c > 0 and check that y(:r) = A sin a3: + B cos as: for some constants A, B solves y\" = a2y. - (5) Use the previous parts to write a solution to the heat equation