A homogeneous second-order linear differential equation, two functions y1 and y2

Question

A homogeneous second-order linear differential equation, two functions y1 and y2 , and a pair of initial conditions are given below. First verify that y1 and y2 are solutions of the differential equation. Then find a particular solution of the form

y = c1y1 + c2y2 that satisfies the given initial conditions.

y'' + 49y = 0;   y1 = cos(7x)    y2 = sin(7x);    y(0) = 10    y(0)=-4
  

y(x)=?

I am struggling on this problem overall, I missed class this week and I am trying to fill in the blanks. Any help would be greatly appreciated.

Explanation / Answer

y1 = cos(7x)    y2 = sin(7x);

y = c1y1 + c2y2

y= c1 cos7x +c2 sin7x

now given inital conditions are

y(0) = 10

y(0) = c1 *1 + c2*0 => C1 = 10

y'(0) = -4

y'(x) = -7 c1 sin7x + 7 c2 cos7x

y'(0) = -4 = 0 + 7c2

c2 = -4/7

solution is

y(x) = 10 cos7x -4/7 sin7x

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