Question

solve y" + 4y = 5 tan(2x)

Answer 1

SOLUTION:-

Given

This is a second order non homogeneous equation.

Its general solution is of the form, where is the general solution of the corresponding homogeneous equation of equation (1)

and is a particular solution of (1).

To find :

Consider equation (2)

Its auxilliary equation is,

i.e,

i.e, m = +i2, -i2

Hence general solution of (2) is,

where A and B are arbitrary constants.

Now to find a particular solution of equation (1):

We may use method of variation of parameters.

Here let and are the two linearly independant solutions of homogeneous equation (1).

Then Wronskian,

Then method of variation of parameters tells that a particular solution is given by,

where r(x) is the RHS of equaiton (2).

Here r(x)=5tan(2x), y1=cos(2x), y2=sin(2x), W=2

--------------(4)

Now we evaluate both the integrals in equation (4) above.

  

Similarly second integral is,

Now putting these integrals in equation (4) we get the particular solution is,

  

Hence the general solution of given non homogeneous differential equation is,

where A and B are arbitrary constants.