Section 4.1 Basics of Differential Equations
References.
Introduction.
Perhaps the most common use of calculus is in describing mathematical and scientific problems as differential equations: equations involving the derivatives of a function. A very simple example is when position \(x\) as a function of time \(t\) is related the the known velocity \(v=f(t)\) by
\begin{equation}
\frac{dx}{dt} = f(t).\tag{4.1.1}
\end{equation}
There are many solutions to this, depending on where the object is initially, so to determine the position function completely, we also need to know the position at one time. For example if at an initial time time \(t=t_0\) the position is known to be \(x_0\text{,}\) we have the initial condition
\begin{equation}
x(t_0) = x_0.\tag{4.1.2}
\end{equation}
General Differential Equations.
We start by defining the main new concept of this Chapter:
Example 4.1.2.
\begin{equation}
\ds \frac{dy}{dx} = \cos x\tag{4.1.3}
\end{equation}
with solutions \(y = F(x) = \sin x + C\) for any constant \(C\text{.}\)
Example 4.1.3.
The above is just integration, with the possible solutions being all anti-derivatives of the right-hand side \(\cos x\text{.}\) More generally, for any continuous function \(f(x)\text{,}\) the differential equation
\begin{equation}
\frac{dy}{dx} = f(x)\tag{4.1.4}
\end{equation}
has the family of solutions consisting of all the antiderivatives of \(f(x)\text{;}\) that is, the general indefinite integral \(\ds y = F(x) = \int f(x)\, dx\text{.}\)
Example 4.1.4.
If we ask for a solution if the above differential equation (4.1.4) with the extra information that \(y(x_0) = y_0\) for some known quantities \(x_0\) and \(y_0\text{,}\) we can work out the constant of integration: there is then a unique solution
\begin{equation*}
y = F(x) = y_0 + \int_{x_0}^x f(t)\, dt
\end{equation*}
Example 4.1.5.
\begin{equation}
\ds \frac{dy}{dx} = y\tag{4.1.5}
\end{equation}
with solutions \(y = F(x) = C e^x\) for any constant \(C\text{.}\)
So there is again a family of solutions with an arbitrary constant, but it is no longer just “\(+ C\)”.
Example 4.1.6.
If we again add the fact that \(y(x_0) = y_0\text{,}\) there is again a unique solution of the differential equation (4.1.5):
\begin{equation*}
y = F(x) = y_0 e^{x-x_0}, = (y_0/e^{x_0}) e^x
\end{equation*}
That is, \(C=y_0/e^{x_0}\text{.}\)
Example 4.1.7.
\begin{equation}
\frac{d^2y}{dt^2} = -g \text{ (a constant)}\tag{4.1.6}
\end{equation}
with solutions
\begin{equation*}
y = F(x) = -\frac{g}{2}t^2 + Ct + D
\end{equation*}
this time with two arbitrary constants, \(C\) and \(D\text{.}\)
Example 4.1.8.
\begin{equation}
\frac{d^2y}{dt^2} = -y\tag{4.1.7}
\end{equation}
with solutions
\begin{equation*}
y = F(t) = C \cos(t) + D \sin(t)
\end{equation*}
again with two arbitrary constants.
Checkpoint 4.1.9.
Verify each of the solutions stated above.
Hint.In each case, compute \(dy/dx\) for the claimed solution (and also \(d^2y/dx^2\) if that appears in the differential equation) and insert the formulas for \(y\text{,}\) \(dy/dx\text{,}\) etc. into the differential equation.
Note that most of the examples above involve on the first derivative, but (4.1.6) and (4.1.7) involve second derivatives. To talk about this feature, we define
Definition 4.1.10.
The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.In this chapter we focus on first order differential equations, which have the general form
\begin{equation}
\frac{dy}{dx} = g(x, y)\tag{4.1.8}
\end{equation}
General and Particular Solutions.
We have seen that a differential equation typically has an infinite collection of solutions, often called its family of solutions.
Furthermore, there is often a single formula describing them all, with the help of one of more constants that can be chosen to get a specific solution like \(Ce^x\) for (4.1.5) or \(\ds -\frac{g}{2}t^2 + Ct + D\) for (4.1.6): such a formula is called a general solution of the differential equation.
For example, the general solution of (4.1.4) is the indefinite integral of the right-hand side function \(f(x)\text{.}\)
Finally, once we have somehow selected one solution from the family (such as \(2e^x\) for (4.1.5) or \(\ds -\frac{g}{2}t^2 + 2t + 3\) for (4.1.6)), this is called a particular solution of the differential equation.
In the case \(dy/dx = g(x)\text{,}\) any one anti-derivative of \(g(x)\) is a particular solution, and finding this is very useful for finding all the solutions: just add an arbitrary constant to get the general solution. Similarly, we will see that it often helps to start by finding any one particular solution, and then get from there to a general solution; however as seen above, this second step is usually not done by just adding a constant!
Initial-Value Problems.
One very common way that a problem involving a differential equation leads to a unique solution is when we also have some information about the value of the function (and maybe of some of its derivatives) at one value of the argument \(x\text{.}\)
Example 4.1.11.
\(\ds \frac{dy}{dx} = y\text{,}\) \(f(0) = 2\) has the unique solution \(y = f(x) = 2 e^x\text{.}\)
Example 4.1.12.
\(\ds \frac{d^2y}{dt^2} = -y\text{,}\) with \(f(0) = 1\text{,}\) \(f'(0) = 0\) has the unique solution \(y = f(t) = \cos t\text{.}\)
The same differential equation with \(f(0) = 0\text{,}\) \(f'(0) = 1\) has the unique solution \(y = f(t) = \sin t\text{.}\)
Checkpoint 4.1.13.
Show that the more general initial value problem
\begin{equation}
\frac{d^2y}{dt^2} = -y,\quad y(0) = y_0,\; y'(0) = y_0'\tag{4.1.9}
\end{equation}
has the particular solution
\begin{equation}
y = y_0 \cos(t) + y_0' \sin(t).\tag{4.1.10}
\end{equation}
In many physical applications, the independent variable (the argument of the solution \(F\) is time; hence the use of argument \(t\) instead of \(x\) in some examples above. Then the extra information is about what we know at one time; often the starting time so that the solution to the differential equation then describes what happens at later times.
Because of this, the combination of a differential equation with such side conditions, like
\begin{equation}
\frac{dy}{dx} = g(x, y), \quad y(x_0) = y_0\tag{4.1.11}
\end{equation}
or (4.1.9) is called an Initial Value Problem.
Study Guide.
Study Calculus Volume 2, Section 4.1 2 ; in particular
- The definitions of differential equation, general solution, particular solution and initial value problem;
- all Examples and Checkpoints;
openstax.org/books/calculus-volume-2/pages/4-1-basics-of-differential-equationsopenstax.org/books/calculus-volume-2/pages/4-1-basics-of-differential-equationswww.desmos.com/calculator
