Wronskian Second Solution/Differential Equations

[tracedinair]

Homework Statement

Given that Φ₂ = Φ₁ * ∫ e^(-∫a(x)dx)) / (Φ₁)^2 dx

and Φ₁ = cos(ln(x)), a = 1/x, solve for Φ₂.

Homework Equations

The Attempt at a Solution

Φ₂ = cos(ln(x)) * ∫ e^(-∫1/x dx)) / cos^(2)(ln(x)) dx

= cos(ln(x)) * ∫ e^(-ln(x)) / cos^(2)(ln(x)) dx

= cos(ln(x)) * - ∫ x / cos^(2)(ln(x)) dx

My problem begins here with trying to solve for that integral. I don't have the slightest idea where to begin, except maybe integration by parts.

 

[djeitnstine]

I believe you have expanded your brackets in the exponential wrong $$e^{-ln(x)}$$ does not equal $$-x$$ but rather $$e^{ln(x^{-1})}$$ which is of course $$\frac{1}{x}$$ in which case a simple u substitution will work

 

[tracedinair]

I believe you have expanded your brackets in the exponential wrong $$e^{-ln(x)}$$ does not equal $$-x$$ but rather $$e^{ln(x^{-1})}$$ which is of course $$\frac{1}{x}$$ in which case a simple u substitution will work

Thanks for catching that mistake.