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An Easy Way to Solve a Simple First Order Differential Equation

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:

Particular solution x(t) = C Homogenous solution x(t) = D exp[-t/A] {D is at this stage unknown}

  • Total solution

x(t) = C + D exp[-t/A]

  • Finding the real solution with initial conditions

Most initial conditions are given for t=0, but it doesn't particularly matter. If we are told that x(T) = Y, sub this in to your total solution, and solve for D.

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?

It 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.