*Express every solution of the given system as the sum of a specific solution plus a solution of the associated homogeneous system:*

**Solution: **This question essentially asks you to use Theorem 3.4. Theorem 3.4 states that to solve the linear system of equations;

(*)

it is sufficient to find *some/ any *(among all the solutions – if one exists) solution , find the solution to the homogeneous system, :

,

and that the general solution to (*) will be .

Realistically you wouldn’t use this method to solve this problem (Q.4 (ii)) – we are more seeing how this theorem works as ye will be using it later in solving linear differential equations.

Namely if

is a suitably linear differential equation, then the general solution may be found by finding *any *particular solution, (by trial and error as much as anything) and adding it to homogeneous solution, (which we know how to find).

Now back to Q. 4 (ii). I don’t know any fool-proof strategy for looking for a particular solution but in this example you might notice that if we add all the equations together, then the second and forth columns all cancel and we are left with:

Now a reasonable guess to make is that for some solution giving us, in this case, that . Throwing this into the third equation:

Throwing this into equation 4, using all these assumptions:

Hence is a potential particular solution. Substituting into the four equations shows that it is indeed a solution.

Now to solve the homogeneous system. Writing in augmented matrix form (with the constant terms all zero):

Apply elementary row operations to get:

Now reading off the solutions (note that #parameters=#variables – rank – so here one parameter. We have the choice because no row is saying that must be something in particular).

Let . Hence . Also . Finally .

Hence using Theorem 3.4, the solution is the sum of the particular and homogeneous solutions:

.

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