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

  • Thread starter digipony
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In summary: State the ODE, please.There might be a mistake somewhere, or your professor really likes Gauss hypergeometric functions.I doubt it; I probably made a mistake.The problem is:Find the general solution by variation of parameters:y'' + y = (tanx)2For 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:yh = C1*e-t+C2Then I needed to find the particular solution which I knew would be in the form u1*y1 + u2*
  • #1
digipony
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Homework Statement



I am in the process of solving second order differential equation by the variation of parameters method. I need to calculate ∫(et)*(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: ∫(et*((sect)2-1)dt
= ∫et*(sect)2 dt -∫(et) dt
=∫et*(sect)2 dt -et
but am stuck here. Any help would be greatly appreciated. Thanks!
 
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  • #2
I think it's not expressible in terms of elementary functions. Find another method for your ODE.
 
  • #4
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.
 
  • #5
State the ODE, please. There might be a mistake somewhere, or your professor really likes Gauss hypergeometric functions.
 
  • #6
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:
yh = 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 = -∫ [itex]\frac{y2*g}{W[y1,y2]}[/itex] dt
and
u2= ∫ [itex]\frac{y1*g}{W[y1,y2 ]}[/itex] dt

(g is (tanx)2, and W is the Wronskian)
 
  • #7
It's not right. y''+y=0 doesn't have the solution you wrote as y_h. No way.
 
  • #8
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 yh= C1*sint + C2*cost
 

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

1. What is the general method for integrating (e^t)*tan^2(t)?

The general method for integrating (e^t)*tan^2(t) is to use integration by parts. This involves choosing one part of the integrand to differentiate and the other part to integrate. This method is often used for integrands that are the product of two functions.

2. Can substitution be used to integrate (e^t)*tan^2(t)?

Yes, substitution can also be used to integrate (e^t)*tan^2(t). The key is to choose the substitution carefully. In this case, the substitution u = tan(t) can be used to simplify the integrand and make it easier to integrate.

3. What is the role of the constant of integration when integrating (e^t)*tan^2(t)?

The constant of integration is an arbitrary constant that is added to the result of an indefinite integral. This is because indefinite integrals are only defined up to a constant. The constant of integration is important because it ensures that the result of the integration is valid for all values of the variable.

4. Are there any special cases to consider when integrating (e^t)*tan^2(t)?

Yes, there are a few special cases to consider when integrating (e^t)*tan^2(t). One is when the integrand has a constant coefficient, which can be factored out and integrated separately. Another is when the integrand has a trigonometric function raised to an even power, which can be rewritten using trigonometric identities.

5. Is there a specific domain for which the integration of (e^t)*tan^2(t) is valid?

The integration of (e^t)*tan^2(t) is valid for all real values of t, as long as the antiderivative exists. However, it is important to note that the integrand is undefined when t = π/2 + nπ, where n is any integer, due to the singularity of the tangent function. In these cases, a different method of integration may be needed.

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