How to integrate (e^t)*tan^2(t)

[digipony]

Homework Statement

I am in the process of solving second order differential equation by the variation of parameters method. I need to calculate ∫(e^t)*(tant)² dt

Homework Equations

(tanx)² = (secx)²-1
∫(secx)dx= lnlsecx*tanxl

The Attempt at a Solution

I have used the trig identity to get this: ∫(e^t*((sect)²-1)dt
= ∫e^t*(sect)² dt -∫(e^t) dt
=∫e^t*(sect)² dt -e^t
but am stuck here. Any help would be greatly appreciated. Thanks!

 

[dextercioby]

I think it's not expressible in terms of elementary functions. Find another method for your ODE.

 

[Dickfore]

I think it's not expressible in terms of elementary functions. Find another method for your ODE.

I think you are right.

http://www.wolframalpha.com/input/?i=Integrate[Exp[t]+Tan[t]^2%2Ct]

 

[digipony]

The problem explicitly asks to solve using variation of parameters. Perhaps I made a mistake in the algebra somewhere before this step; I will go to my professor. Thanks for your input, now I won't waste a hour or two staring at and trying to solve this integral.

 

[dextercioby]

State the ODE, please. There might be a mistake somewhere, or your professor really likes Gauss hypergeometric functions.

 

[digipony]

I doubt it; I probably made a mistake. Here is the problem:

Find the general solution by variation of parameters:
y'' + y = (tanx)²

For these, I know you find the solution by finding and adding together the solution to the homogeneous equation with that of your particular solution.

I found the solution to the homogeneous equation to be:
y_{h =} C₁*e^-t+C₂

Then I needed to find the particular solution which I knew would be in the form u₁*y₁ + u₂*y₂
where
u₁ = -∫ \frac{y₂*g}{W[y₁,y₂]} dt
and
u₂= ∫ \frac{y₁*g}{W[y₁,y₂ ]} dt

(g is (tanx)², and W is the Wronskian)

 

[dextercioby]

It's not right. y''+y=0 doesn't have the solution you wrote as y_h. No way.

 

[digipony]

Ah! I see. I was using reduction of power method and messed up. I had gotten r² + r =0 whereas it should be r²+1=0 . Therefore I should get an imaginary root, and the solution is y_h= C₁*sint + C₂*cost