Check whether the function yequalsStartFraction cosine 2 x Over x EndFraction is a solution of x y prime plus yequalsnegative 2 sine 2 x with the initial condition y (StartFraction pi Over 4 EndFraction) equals0.

The question is:

Check whether the function:

y = [cos (2x) ]/x

is a solution of

xy' + y = - 2sin (2x)

with the initial condition y (π/4) = 0

Answer:

To check if the function y = [cos (2x) ]/x is a solution of the differential equation xy' + y = - 2sin (2x), we need to substitute the value of y and the value of the derivative of y on the left hand side of the differential equation and see if we obtain the right hand side of the equation.

Let us do that.

y = [cos (2x) ]/x

y' = (-1/x²) [cos (2x) ] - (2/x) [sin (2x) ]

Now,

xy' + y = x{ (-1/x²) [cos (2x) ] - (2/x) [sin (2x) ]} + ([cos (2x) ]/x

= (-1/x) cos (2x) - 2sin (2x) + (1/x) cos (2x)

= - 2sin (2x)

Which is the right hand side of the differential equation.

Hence, y is a solution to the differential equation.