Solving the differential equation f'(x) + ((1-2x)/x²)y = 1

[Cyannaca]

Hi, I would need help on this problem. I have to solve this equation
f'(x) + ((1 -2x)/x^2)*y = 1. I started the problem and I was stuck at
y= (x^2)*e^(1/x) * integral (e^(-1/x))/(x^2) . I have no idea how to do this so if anyone knows how, it would be really appreciated.

 

[Hurkyl]

By "do this" I presume you mean you are having trouble finding an elementary expression for

$$\int \frac{e^{-1/x}}{x^2} \, dx$$

What techniques have you tried to apply?

 

[marlon]

Try a substitution , realizing that (1/x²)dx = - d(1/x)

regards
marlon

 

[Cyannaca]

(1/x²)dx = - d(1/x). Thanks a lot, I was quite lost (trying to integrate by parts...)but I finally solved it. Thanks.

 

[maverick280857]

For integration by parts, try ILATE (I = Inverse Trig, L = Logarithmic, A = Algebraic, T = Trig, E = Exponential). This pretty much sums up the order in which the first function should be chosen if you wish to integrate something of the form,

$$\int udv = uv - \int v du$$

(ie--how to chose u).

Adios
Vivek

 

[Zurtex]

Erm, have you thought about what the derivative of:

$$e^{\frac{-1}{x}}$$

is? I think that will solve your problem

Edit: Sorry, missed that you had solved it.