First Order Differential Equations Involving Vectors

The simplest first order differential equation is of the form(1). We can solve this equation by assuming a solution of the formso that Substituting these into the differential equation (1) gives

since that would mean the solution is trivial (and does not satisfy any boundary conditions) andso


If instead we have a first order differential equation with vectors, say(2) withwhenwe can still assume a solution of the formbut nowis also a vectorwhich we have to find given the initial conditions.

IfthenSubstituting into the differential equation (2) gives

As before, we can factorise withto give

Again as beforeandsohence(3)

To finduse the initial conditionswhenhenceso

We can write this solution in vector form as

More complicated equations may include constant terms added. For example(4)


We find the solution in two parts. The first part will be the solution toThis is (3) above. The second part is any solution to (4). We can see that ifis a constant thenso putinto (4) then

The general solution is the sum ofandsoWe can find using the initial conditionswhen


We can write this as

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