Basic concepts · Definition

Differential equation

differential equationsfundamentals

Almost anything that changes over time is described by saying how it changes, not what it equals; that description is a differential equation.

Definition

A differential equation is an equation \(F=0\) whose unknown is a function \(y\) and which relates it to one or more of its derivatives.

\[F\left(x,\,y,\,y',\,\dots,\,y^{(n)}\right)=0\]

In algebra, solving \(x^2=4\) means finding numbers. Here the unknown is a function: we look for a \(y(x)\) that, once differentiated and substituted, makes the equality true. For example, \(y'=2y\) asks for a function whose derivative is twice itself, and \(y=e^{2x}\) works.

When the function depends on a single variable, the derivatives are ordinary and we speak of an ordinary differential equation (ODE), the subject of this course. If the function depends on several variables and partial derivatives appear, as in \(u_t=u_{xx}\), the equation is a partial differential equation and needs other tools.

Differential equations arise from modeling: the rate of change of a population, the cooling speed of a body or the acceleration of a mass on a spring are derivatives, and the physical law equates them to an expression of the quantity itself. Setting up the equation is half the work; solving it is the other half.

A common mistake is to confuse a differential equation with a derivative formula: \(y'=2x\) is not «the derivative of \(y\)» but a condition \(y\) must satisfy. Next we classify these equations by order and linearity, and then see what solving them means.

Think about it

Is \(y=x^2\) a differential equation? What about \(y'=2x\)?

The first is not: it has no derivatives, it is a function. The second is: it relates \(y\) to its derivative.

Exercises 10

  1. Which of the following is a differential equation?
  2. In \(\dfrac{dy}{dt}=-0.5\,y\), which is the independent variable?
  3. Differentiating \(y=Ce^{3x}\) and eliminating the constant \(C\) gives the equation:
  4. Which of the following is NOT a differential equation?
  5. Differentiating \(y=x^2+C\) and eliminating \(C\) gives:
  6. Which of the following differential equations involves partial derivatives?
  7. Which function satisfies the equation \(y''+y=0\)?
  8. If the rate of change of a population \(P\) with respect to time is proportional to \(P\), the model is:
  9. Solving \(y'=2y\) means finding:
  10. In \(\dfrac{dy}{dx}=x+y\), which statement is correct?

References

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §1.3
  • Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §1.1