Our task is thereforce to solve this equation. The most important step is to pick which set of equations you want to deal with first. The obivous choice is

reason being that using the  differential will give us a mismatch of terms, we can’t integrate either x w.r.t z or z w.r.t y, knowing that x, y, and z could each be expressed in terms of x, y, or z.

Separating the variables and integrating

My way of dealing with the arbituary constants is to group them on one side and that equate them to a final arbituary constant to simplify the algebra.

We shall now deal with the equation

as we have expressed x in terms of z so integrating w.r.t to z is now possible.

On separating the varaibles and integrating,

Please note that the  is important because when we integrate it w.r.t z, it will become the coefficient of a z term.

where  is the final arbituary constant we will use.

While you can choose to express the solutions in either x, y and z, we have chosen to express them in z simply because z is raise to certain powers, in this case 4 and 5. This is much neater as expressing the solutions in another variable may lead to taking fraction powers. It just goes to show it pays to express the solution in the ideal variable.

Thus our general solution of the differential equestion is

For our paricular solution, we are concern with the line of force that passes through the point . We substitute the values  into the equations to find  and  which in this case is

Remember earlier in this chapter we said that there are an infinite amount of lines of force following this vector field. Well, this is exactly what the general solution tells us. To find that particular line of force, we substitute the x,y,z coordinates of that point.