# general solution to linear first order differential equations

• Feb 4th 2009, 08:18 PM
morrobay
general solution to linear first order differential equations
Hello,
What is the basis for e, the natural log, as a general solution in D.E. ?
For example:

x' (t) + 2x (t) =6
x(t) = Ce^-2x +3

I understand d/dx (e^x) =e^x
In general solutions what is the concept and analysis of the natural log
application ?
• Feb 4th 2009, 08:23 PM
ThePerfectHacker
Quote:

Originally Posted by morrobay
Hello,
What is the basis for e, the natural log, as a general solution in D.E. ?
For example:

x' (t) + 2x (t) =6
x(t) = Ce^-2x +3

I understand d/dx (e^x) =e^x
In general solutions what is the concept and analysis of the natural log
application ?

You have $\displaystyle x' + 2x = 6$ multiply both sides by $\displaystyle e^{2t}$ and so $\displaystyle e^{2t}x' + 2e^{2t}x = 6$.
This can be written as $\displaystyle \left( e^{2t} x \right)' = 6$.
Can you continue?
• Feb 4th 2009, 08:43 PM
morrobay
Thanks,
Actually my question is on why (e) the natural log is used in solving D.E
That is, what is the basis or analysis of why (e) is used in the general solutions ?
• Feb 4th 2009, 08:58 PM
ThePerfectHacker
Quote:

Originally Posted by morrobay
Thanks,
Actually my question is on why (e) the natural log is used in solving D.E
That is, what is the basis or analysis of why (e) is used in the general solutions ?

Because $\displaystyle e^t$ has the special property that its it is own derivative.
Look at how this fact was used in the solution of the differencial equation.