§1 Introduction to Differential Equations

A differential equation, or DE, is an equation involving an unknown function and one or more of its derivatives. Instead of solving for a number, we are looking for a function whose derivatives satisfy the equation.

For example, let $t$ be the independent variable, let $N(t)$ be the unknown function, and let $r$ be a constant. The prime means differentiation with respect to $t$:

$$N'(t)=\frac{dN}{dt}.$$

Now consider the differential equation

$$N'(t)=rN(t)$$

It says that the rate of change $N'(t)$ is proportional to the current value $N(t)$.

Differential equations as models

Differential equations are useful because many physical rules describe how something changes rather than giving its value directly.

Example: a falling object

Let $v(t)$ be the downward velocity of an object at time $t$. Let $m$ be the object's mass, $g$ the acceleration due to gravity, and $\gamma$ the drag coefficient. Newton's second law says

$$\text{mass}\times\text{acceleration}=\text{sum of forces}.$$

Since acceleration is $v'(t)$, the left side is $mv'(t)$. If downward is positive, gravity contributes $mg$, while drag acts upward with magnitude $\gamma v(t)$. Therefore,

$$mv'(t)=mg-\gamma v(t).$$

The equation balances two effects: gravity speeds up the fall, while drag opposes the motion.

Example: population growth

Let $N(t)$ be the size of a population at time $t$. In an idealized environment with unlimited resources, assume that the population's rate of change is proportional to its current size. If $r$ is the proportionality constant, then

$$N'(t)=rN(t).$$

For a positive population, divide by $N(t)$ and use the chain rule. Here, $\ln$ is the natural logarithm:

$$\frac{N'(t)}{N(t)}=r, \qquad \frac{d}{dt}\ln N(t)=r.$$

Integrating with respect to $t$ introduces a constant of integration, which we call $C$:

$$\ln N(t)=rt+C.$$

Exponentiating both sides,

$$N(t)=e^{rt+C}=e^Ce^{rt}.$$

If we write $k=e^C$, then

$$N(t)=ke^{rt},\qquad k>0.$$

The constant $k$ is the initial population because

$$N(0)=ke^0=k.$$

Different values of $k$ give different solutions. For a population model, $k>0$. The sign of $r$ determines what happens over time:

• $r>0$: exponential growth;
• $r=0$: constant population;
• $r<0$: exponential decay.

The unlimited-resource assumption is only an idealization. It is useful over limited time periods, but real populations cannot grow exponentially forever.

Ordinary and partial differential equations

An ordinary differential equation, or ODE, uses derivatives with respect to one independent variable. These are all ODEs:

$$N'(t)=rN(t),$$ $$y''(x)+y(x)=0,$$ $$mv'(t)=mg-\gamma v(t).$$

A partial differential equation, or PDE, involves partial derivatives of a function that depends on more than one independent variable. A partial derivative uses the symbol $\partial$ and measures change with respect to one variable while the others are held fixed.

For example, let $u(x,t)$ represent temperature at position $x$ and time $t$. The one-dimensional heat equation can be written as

$$\frac{\partial u}{\partial t} = \frac{\partial^2u}{\partial x^2}.$$

The unknown function $u(x,t)$ depends on two independent variables, $x$ and $t$.

The order of an ODE

The order of an ODE is the number of times the unknown function is differentiated in the highest derivative that appears.

The exponent on $y$ does not determine the order. For example, the $y^2$ term in the last equation makes the equation nonlinear, but it does not make it second order. The fourth derivative $y^{(4)}$ determines the order.

The independent variable is often omitted when it is clear from context. For example,

$$y^{(4)}+y''+y^2=0$$

is shorthand for an equation involving $y(t)$, $y''(t)$, and $y^{(4)}(t)$.

For any positive integer $k$, the notation

$$y^{(k)}(t)=\frac{d^ky}{dt^k}(t)$$

means the $k$th derivative of $y$ with respect to $t$.

A general form for an ODE

Let $F$ be a function that combines the independent variable, the unknown function, and its derivatives up to order $n$. An $n$th-order ODE can be written implicitly as

$$F\bigl(t,y(t),y'(t),\ldots,y^{(n)}(t)\bigr)=0.$$

The function $F$ returns a real number. The symbol $\mathbb R$ means the set of real numbers. In symbols,

$$F:\mathbb R^{n+2}\to\mathbb R.$$

There are $n+2$ inputs because they are

$$t,\ y(t),\ y'(t),\ldots,y^{(n)}(t).$$

Example: building an ODE from $F$

Suppose

$$F(x,y,z)=x^2+2xy+z^3.$$

Substituting $x=t$, $y=y(t)$, and $z=y'(t)$ gives

$$F\bigl(t,y(t),y'(t)\bigr) =t^2+2t\,y(t)+\bigl(y'(t)\bigr)^3.$$

Therefore,

$$t^2+2t\,y(t)+\bigl(y'(t)\bigr)^3=0$$

is an implicit first-order ODE. It is first order because $y'$ is the highest derivative present, even though $y'$ is cubed.

What counts as a solution?

Let $I=(a,b)$ be the open interval of values strictly between $a$ and $b$. The notation $y:I\to\mathbb R$ says that $y$ takes an input from $I$ and returns a real number. The symbol $\in$ means “is an element of,” so $t\in I$ means that $t$ belongs to the interval $I$.

A solution of

$$F\bigl(t,y(t),y'(t),\ldots,y^{(n)}(t)\bigr)=0$$

on $I$ is a function $y:I\to\mathbb R$ that has all the derivatives needed by the equation and makes the equation true for every $t\in I$.

Example: checking a proposed solution

Consider

$$y''+y=0.$$

To check whether $y(t)=\cos t$ is a solution, differentiate it:

$$y'(t)=-\sin t, \qquad y''(t)=-\cos t.$$

Now substitute into the ODE:

$$y''(t)+y(t) =-\cos t+\cos t =0.$$

The equation is true for every real $t$, so $y(t)=\cos t$ is a solution on $\mathbb R$.

Linear and nonlinear ODEs

An $n$th-order ODE is linear when $y$ and its derivatives appear only to the first power, are not multiplied together, and are not placed inside nonlinear functions. Its coefficients may depend on the independent variable $t$.

Let $a_0(t),a_1(t),\ldots,a_n(t)$ be coefficient functions, and let $h(t)$ be the right-hand side. A general $n$th-order linear ODE has the form

$$a_0(t)y^{(n)}(t) +a_1(t)y^{(n-1)}(t) +\cdots +a_n(t)y(t) =h(t),$$

The coefficients and the right-hand side may all depend on $t$ without making the equation nonlinear.

Example: a linear ODE

$$ty''(t)+\cos(t)y'(t)+t^3y(t)=\sin t$$

is a second-order linear ODE. Matching it to the general form gives

$$n=2, \qquad a_0(t)=t, \qquad a_1(t)=\cos t, \qquad a_2(t)=t^3, \qquad h(t)=\sin t.$$

Example: a nonlinear ODE

$$y'(t)+\bigl(y(t)\bigr)^2=\sin t$$

is nonlinear because $y(t)$ is squared. The equation is still first order because $y'$ is its highest derivative.

General solutions

A general solution is a family containing all the solutions of an ODE under consideration. The members of the family are distinguished by one or more arbitrary constants.

For example, the general real-valued solution of

$$y'(t)=2y(t)$$

is

Since $\mathbb R$ is the set of real numbers, we write the arbitrary constant as

$$y(t)=ke^{2t},\qquad k\in\mathbb R.$$

This says that $k$ may be any real number.

We can verify this directly:

$$y'(t)=2ke^{2t}=2y(t).$$

Every real value of $k$ gives a solution:

• $k>0$ gives a positive exponential curve;
• $k=0$ gives the zero solution;
• $k<0$ gives a negative exponential curve.

Without an additional condition, such as a known value of $y$ at one time, the ODE therefore has infinitely many solutions.

Scalar equations and systems

A scalar ODE has one unknown scalar-valued function. A system of ODEs is a collection of differential equations that must be satisfied together, usually involving several unknown functions.

For example, let $x(t)$ and $y(t)$ be two unknown functions. Then

$$\begin{cases} x'(t)=x(t)y(t)+\cos t,\\ y'(t)=x(t)+\bigl(y(t)\bigr)^2 \end{cases}$$

is a system for the two unknown functions $x(t)$ and $y(t)$. The equations are coupled because each one depends on the other unknown function.

Autonomous and non-autonomous ODEs

An ODE is autonomous when the independent variable does not appear explicitly in the equation. For example,

$$N'=rN, \qquad v'=g-\frac{\gamma}{m}v, \qquad y''+y=0$$

are autonomous when $r$, $g$, $\gamma$, and $m$ are constants. Their solutions still depend on time; autonomous only means that time does not appear explicitly in the equation itself.

An ODE is non-autonomous when the independent variable does appear explicitly. For example,

$$t^2y'(t)+\cos(t)y(t)=0$$

is non-autonomous because $t$ appears in $t^2$ and $\cos t$.