(a) For what values of k does the function y = cos(kt) satisfy the differential equation 25y" = -9y? (Enter your answers as a comma-separated list.) 3 3 5'5 -(--) k = (b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution. We begin by calculating the following. y = A sin(kt) + B cos(kt) = y' = Ak cos(kt) - Bk sin(kt) => y-Aksin (kt)- Bk2 cos(kt) Note that the given differential equation 25y" = -9y is equivalent to 25y" + 9y= 0 Now, substituting the expressions for y and y" above and simplifying, we have = = LHS 25y+9y 25-Ak²sin (kt) -Ak²sin (kt) - Bk² cos(kt) +9(A sin(kt) + B cos(kt)) --25 Aksin (kt) 25Bk2 cos(kt)+9A sin(kt) + 9B cos(kt) = - (9 25k2) sin(kr) )+ +(925k2) B cos(kt) = 0 since for all value of k found above, k² = 9 = 25

Solve the Following differential equations:

Q-1. Form the first-order differential equation of y = a sin 0 – bcos 0, x = a cos 0+b sin 0

-3t y = 3 sin(t) – cos(t) + is a solution to the differential equation 3. Show that : y'+ 3 y = 10 sin(t)

Please help me answer. This is Differential Equations.