Verify if y=c*sin2x+3*cos 2x is the general solution for d^2y/dx^2+4y=0?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The function y = c*sin2x+3*cos 2x

dy/dx = c*(cos 2x)*2 + 3*(-sin 2x)*2

d^2y/dx^2 = 2c*2(-sin 2x) - 6*2(cos 2x)

=> -4c*sin 2x - 12*cos 2x

d^2y/dx^2 + 4y

=> -4c*sin 2x - 12*cos 2x + 4(c*sin 2x + 3*cos 2x)

=> -4c*sin 2x - 12*cos 2x + 4c*sin 2x + 12*cos 2x

=> 0

The value of d^2y/dx^2 + 4y = 0

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll have to determine the first derivative of the function:

y' = 2c*cos 2x - 6*sin 2x

Now, we'll determine the second derivative of y:

y" = -4c*sin 2x - 12*cos 2x

Now, we'll substitute y" and y into the given differential equation to verify if it is cancelling.

d^2y/dx^2 + 4y = (-4c*sin 2x - 12*cos 2x) + 4(c*sin 2x + 3*cos 2x)

We'll remove the brackets:

d^2y/dx^2 + 4y = -4c*sin 2x - 12*cos 2x + 4c*sin 2x + 12*cos 2x

We notice that we can eliminate all terms from the right side:

d^2y/dx^2 + 4y = 0

Therefore, y = c*sin 2x + 3*cos 2x represents the general solution of the equation d^2y/dx^2 + 4y.

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