Introduction to Differential Equations

Often we do not have just one dependent variable and one equation. And as we will see, we may end up with systems of several equations and several dependent variables even if we start with a single equation.

We usually also have an initial condition. Just like for single equations we specify \(x_1\text{,}\) \(x_2\text{,}\) and \(x_3\) for some fixed \(t\text{.}\) For example, \(x_1(0) = a_1\text{,}\) \(x_2(0) = a_2\text{,}\) \(x_3(0) = a_3\text{.}\) For some constants \(a_1\text{,}\) \(a_2\text{,}\) and \(a_3\text{.}\) For the second order system we would also specify the first derivatives at a point. And if we find a solution with constants in it, where by solving for the constants we find a solution for any initial condition, we call this solution the general solution. Best to look at a simple example.

Generally, we will not be so lucky to be able to solve for each variable separately as in the example above, and we will have to solve for all variables at once. While we won’t generally be able to solve for one variable and then the next, we will try to salvage as much as possible from this technique. It will turn out that in a certain sense we will still (try to) solve a bunch of single equations and put their solutions together. Let’s not worry right now about how to solve systems yet.

Let us consider some simple applications of systems and how to set up the equations.

A similar process can be followed for a system of higher order differential equations. For example, a system of \(k\) differential equations in \(k\) unknowns, all of order \(n\text{,}\) can be transformed into a first order system of \(n \times k\) equations and \(n \times k\) unknowns.

It is useful to go back and forth between systems and higher order equations for other reasons. For example, software for solving ODE numerically (approximation) is generally for first order systems. To use it, you take whatever ODE you want to solve and convert it to a first order system. It is not very hard to adapt computer code for the Euler or Runge–Kutta method for first order equations to handle first order systems. We simply treat the dependent variable not as a number but as a vector. In many mathematical computer languages there is almost no distinction in syntax.

A system where the equations do not depend on the independent variable is called an autonomous system. For example the system \(y'=2y-x\text{,}\) \(y'=x\) is autonomous as \(t\) is the independent variable but does not appear in the equations.

For autonomous systems we can draw the so-called direction field or vector field, a plot similar to a slope field, but instead of giving a slope at each point, we give a direction (and a magnitude). The previous example, \(x' = 2y-x\text{,}\) \(y' = x\text{,}\) says that at the point \((x,y)\) the direction in which we should travel to satisfy the equations should be the direction of the vector \(( 2y-x, x )\) with the speed equal to the magnitude of this vector. So we draw the vector \((2y-x,x)\) at the point \((x,y)\) and we do this for many points on the \(xy\)-plane. For example, at the point \((1,2)\) we draw the vector \(\bigl(2(2)-1,1\bigr) = (3,1)\text{,}\) a vector pointing to the right and a little bit up, while at the point \((2,1)\) we draw the vector \(\bigl(2(1)-2,2\bigr) = (0,2)\) a vector that points straight up. When drawing the vectors, we will scale down their size to fit many of them on the same direction field. If we drew the arrows at the actual size, the diagram would be a jumbled mess once you would draw more than a couple of arrows. So we scale them all so that not even the longest one interferes with the others. We are mostly interested in their direction and relative size. See Figure 7.2.

We can draw a path of the solution in the plane. Suppose the solution is given by \(x = f(t)\text{,}\) \(y=g(t)\text{.}\) We pick an interval of \(t\) (say \(0 \leq t \leq 2\) for our example) and plot all the points \(\bigl(f(t),g(t)\bigr)\) for \(t\) in the selected range. The resulting picture is called the phase portrait (or phase plane portrait). The particular curve obtained is called the trajectory or solution curve. See an example plot in Figure 7.3. In the figure the solution starts at \((1,0)\) and travels along the vector field for a distance of 2 units of \(t\text{.}\) We solved this system precisely, so we compute \(x(2)\) and \(y(2)\) to find \(x(2) \approx 2.475\) and \(y(2) \approx 2.457\text{.}\) This point corresponds to the top right end of the plotted solution curve in the figure.

Notice the similarity to the diagrams we drew for autonomous systems in one dimension. But note how much more complicated things become when we allow just one extra dimension.

We can draw phase portraits and trajectories in the \(xy\)-plane even if the system is not autonomous. In this case, however, we cannot draw the direction field, since the field changes as \(t\) changes. For each \(t\) we would get a different direction field.

That is, a unique solution exists for any initial condition given that the system is reasonable (\(F_j\) and its partial derivatives in the \(x\) variables are continuous). As for single equations we may not have a solution for all time \(t\text{,}\) but at least for some short period of time.

As we can change any \(n\)th order ODE into a first order system, then we notice that this theorem provides also the existence and uniqueness of solutions for higher order equations that we have until now not stated explicitly.