Find the of two solutions of the differential equation witho

Find the of two solutions of the differential equation without solving the equation. r^2 y\" -t(t + 2)y\' + (t + 2)y = 0 Find the solution of the given initial value problem sketch the graph of the solution and describe its behavior for increasing t. y\" + 44 = 0 y(0) = 0 y\'(0) = 1

Solution

y\"+4y=0

auxiliary equation is

(D^2+4)y=0

D^2=-4

D=+2i,-2i

complimentary function is y(x)=Acos2x+Bsin2x

y(0)=Acos0+Bsin0=A=0

y\'(x)=-Asin2x+Bcos2x

y\'(0)=B=1

hence rquired solution is y(x)= sin2x