Solve the following differential equation with the given bou

Solve the following differential equation with the given boundary conditions.

- If there are infinitely many solutions, use c for any undetermined constants. -

If there are no solutions, write No Solution.

- Write answers as functions of x (i.e. y=y(x)).

y+9y=0

A) Boundary conditions:

y(0)=1

y(/3)=1

y=--------------------------------

B)Boundary conditions:

y(0)=-1

y(19/6)=1

y=-------------------------------------

C)Boundary conditions:

y(0)=1

y(/3)=1

y=-----------------------------------------

y+9y=0

First solve this De and find general solution

Aux equation is m^2+9 =0

m = 3i , -3i

Hence solution is y = A cos 3x + B sin 3x

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a) Boundary conditions:

y(0)=1

y(/3)=1

-1 = A and 1 = 1+0

Hence B can be artbitrary

y(x) = -1 cos 3x + B sin 3x

--------------------------------------------

y(0)=-1

y(19/6)=1

Substitute to get

-1 = A

and 1 = A cos 19pi/2 + B sin 19pi/2

i.e. 1 = -1(0) + B(-1)

B = -1

Hence y(x) = -cos 3x - sin 3x

-------------------------------

c) y(0)=1

y(/3)=1

-1 = A and

-1 = -1 cos pi + 0

Hence no solution for A and B