First order homogenous DE with variable coefficient

[MrGandalf]

I was stumped by this differential equation. The function x = x(t).

$$x^{\tiny\prime\prime} + \frac{1}{t}x^{\tiny\prime} = 0$$.

You have the initial values x(a) = 0 and x(1) = 1.

What I did was to introduce a new function u = x', so I ended up with the first order homogenous DE:
$$u^{\tiny\prime} + \frac{1}{t}u = 0$$.

This can't be that difficult, but the book which I am using does not give any details how to solve this particular problem. I only know how to solve it if I have a constant, or a function of t, on the right side.

The solution to the problem is:
$$x(t) = 1 - \frac{\log t}{\log a}$$.

Hope someone can shed some light on this. Thank ye, ye scurvy landlubber! Yarrr!

 

[AiRAVATA]

When you did a reduction of order, you arrived to a separable equation that isn't difficult to solve at all. There is no need to reduce order since the original equation can be rewritten in a simpler form. What happens if you multiply the second order ode by t?. Can you simplify it (i.e. the derivative of something)?

 

[MrGandalf]

Yes! Of course! How could I miss that? :)

Thank you for helping me!