## 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 form so that Substituting these into the differential equation (1) gives  since that would mean the solution is trivial (and does not satisfy any boundary conditions) and so If when then If instead we have a first order differential equation with vectors, say (2) with when we can still assume a solution of the form but now is also a vector which we have to find given the initial conditions.

If then Substituting into the differential equation (2) gives As before, we can factorise with to give Again as before and so hence (3)

To find use the initial conditions when hence so We can write this solution in vector form as More complicated equations may include constant terms added. For example (4)

with when We find the solution in two parts. The first part will be the solution to This is (3) above. The second part is any solution to (4). We can see that if is a constant then so put into (4) then The general solution is the sum of and so We can find using the initial conditions when  Hence We can write this as 