Given the second order linear homogeneous differential equat

Given the second order linear homogeneous differential equation y\'\'+25y=0 and the functions y_1(x)=cos5x , y_2(x)=sin5x. y_1(x) and y_2(x) for a fundamental set of solutions to the linear homogeneous differential equation.

Find the general solution.

Let y(0)=2 and y\'(0)=-4. Find the solution to the initial value problem.

Solution

y\"+25y =0 has auxialary equation as

m^2+25 =0 Hence roots are m = 5i, -5i

So y1 = cos5x and y2 = sin 5x

and general solution is

y = c1 cos 5x + c2 sin 5x

y(0)=2 and y\'(0)=-4

Plug in initial values

y\' = -5c1 sin 5x +5c2 cos 5x

y (0) = c1(1)+0 = 2

i.e. c1 =2

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y\'(0) = -4

i.e. 0+5c2(1) = -4

or c2 = -4/5

Hence solution is

y = 5 cos 5x -4/5 sin 5x