Solving Differential Equations: Understanding the Steps

[kingpen123]

I am looking for help solving these two differential equations:

1. x'=-x

2. x'=x², x(0)=x₀

The solutions are x(t)=e^-tx₀, and x(t)=x₀/(1-x₀t).

I just don't understand what steps were being done to get those solutions. If someone could point me in the right starting point or show me some steps to get these solutions it would be much appreciated.

 

[CompuChip]

Hi Kingpen,

Can you maybe show us some more information or for example point us to your textbook? The problem is that there are many ways to "solve" these equations. In particular, you can "guess" the solution and then show that it works and you can even use the differential equation x' = x with initial value x(0) = 1 as the definition of the function f(t) = e^t. So knowing by which approach / on which level you would like to solve these equations may help.