## 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 = 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.