Aren't differential equations annoying? They can be quite confusing and incredibly difficult to solve, even the simple first order ones!

Well here's a simple way to solve simple first order differential equations:

Take the example (I don't know how to use LaTex, so excuse the dodgy-ness)

a x'(t) + b x(t) = c Initial condition: x[T] = Y {some time T}

Rearrange to obtain A x'(t) + x(t) = C (ie coefficient of x(t) = 1)

Differential equations have a particular and homogenous solution which are added together to form a total solution. In this case:

Total solution

Finding the real solution with initial conditions

This will give D = (Y - C) / exp[-T/A]

For T=0, this gives D = Y - C

Thus, a general solution to A x'(t) + x(t) = C Initial condition: x[T] = Y {any T}

is x(t) = C + (Y-C / exp[-T/A]) exp[-t/A]

= C + (Y-C)exp[(T-t)/A] for T=0 x(t) = C + (Y-C)exp(-t/A)

ie 6x'[t]+3x[t]=18; x[0] = 14

–> 2x'[t]+x[t]=6

–> x[t] = 6 + 8exp[-t/2]

This is particularly useful for solving LR and RC circuits in electrical engineering, which is where my inspiration for this came from.

## Comments

## method of undetermined coefficients?

Wed, 22/04/2009 - 07:28 — jaspercrowneIt would be better to give this article a more specific name, and place it into a hierarchy under the Differential equations front page. In particular "simple" is not a well-defined term here. Presumably "linear" is what is meant, and furthermore a better discussion of linearity should be given, in order to motivate the separation into "homogeneous" and "particular" solutions.

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