Particular Solution of an Inhomogeneous Second Order ODE

[0131413]

Homework Statement

A particular solution of y'' + 4y = tanx

Answer choices are:

(a) 1/2*cos(2x)ln|sec(2x)+tan(2x)|
(b) -1/2*cos(2x)ln|sec(2x)+tan(2x)|
(c) 1/2*sin(2x)(ln*cos(x)+x*sec(2x))
(d) 1/2*sin(2x)(ln*cos(x)-x*sec(2x))
(e) none of the above

Homework Equations

The Attempt at a Solution

I used variation of parameters with y₁=cos(x) and y₂=sin(x)

Wronskian(cos(x) sin(x))=1

y_p=-v₁y₁+v₂y₂

v₁'=sin(x)tan(x)
=(sin(x))^2/cos(x)
=(1-(cos(x))^2)/cos(x)
=1/cos(x)-cos(x)

v₁=ln|tan(x)+sec(x)|-sin(x)

v₂'=cos(x)tan(x)
=sin(x)

v₂=-cos(x)

y_p=-cos(x)(ln|tan(x)+sec(x)|-sin(x))+sin(x)(-cos(x))
=-cos(x)ln|tan(x)+sec(x)|+cos(x)sin(x)-sin(x)cos(x)
=-cos(x)ln|tan(x)+sec(x)|

Given the answer choices, now I feel like I've missed something fundamental. It's probably obvious. So sorry.

 

[Metaleer]

The solutions to the homogeneous part are not correct. Think cos(2x) and sin(2x).

 

[SVXX]

Why not use the normal method?
Rewriting in terms of the D operator,
(D² + 4)y = tan(x).

The auxiliary equation of this D.E. is D = ±2i, henceforth we obtain the complementary function to be -:

CF = e⁰[c₁cos2x + c₂sin2x]

For a function such as tan(x), the particular integral is given by -:

Operating separately, we have -:

and

Once you compute these integrals, the complete solution of the differential equation will be given by -:

CS = CF + PI

 

[0131413]

The solutions to the homogeneous part are not correct. Think cos(2x) and sin(2x).

Why not use the normal method?
Rewriting in terms of the D operator,
(D² + 4)y = tan(x).

The auxiliary equation of this D.E. is D = ±2i, henceforth we obtain the complementary function to be -:

CF = e⁰[c₁cos2x + c₂sin2x]

For a function such as tan(x), the particular integral is given by -:

Operating separately, we have -:

and

Once you compute these integrals, the complete solution of the differential equation will be given by -:

CS = CF + PI

Thanks a lot. I understand now. I was not sure about the fundamental set when dealing with tan(x), but I see it now.

SVXX: Because the lesson was on variation of parameters.