Basic concepts · Definition

Order and linearity

differential equationsclassification

Before you start you need to know

Before trying to solve a differential equation it pays to classify it: its order and linearity decide which method applies.

Definition

The order of a differential equation is that of its highest derivative; the equation is linear if the unknown and its derivatives appear only to the first power and are never multiplied together.

\[a_n(x)\,y^{(n)}+\cdots+a_1(x)\,y'+a_0(x)\,y=g(x)\]

The order can be read directly: it is the number of the highest derivative that appears. The equation \(y'''+2y'=x\) is third order even though it also has a first derivative. Exponents do not change the order: \((y'')^2+y'=x\) is still second order.

Linearity is a condition on how the unknown enters. The equation is linear when \(y\) and its derivatives appear only multiplied by functions of \(x\) and added, as in the formula above. Products such as \(y\,y'\), powers such as \(y^2\) or \((y')^3\), and functions of the unknown such as \(\sin y\) or \(e^{y}\) make it nonlinear.

Coefficients depending on \(x\) do not break linearity: \(x^2y''+e^xy'+y=0\) is linear. What matters is dependence on \(y\). That is why \(y'+y=y^n\) is linear only for \(n=0\) and \(n=1\).

Linear equations have a complete theory and systematic methods, while nonlinear ones often need special tricks or approximations. Many nonlinear models are therefore linearized near a point: the pendulum \(\theta''+\frac{g}{L}\sin\theta=0\) becomes \(\theta''+\frac{g}{L}\theta=0\) for small angles.

Think about it

Does raising \(y\) to the fifth power in \(y''=y^5\) change the order? Is it linear?

The order stays 2. It is not linear because of the \(y^5\) term.

Is \(y'=x\,y+\sin x\) linear?

Yes: \(y'-xy=\sin x\) has the linear form with \(a_1=1\), \(a_0=-x\) and \(g(x)=\sin x\).

Exercises 10

  1. What is the order of \(y'''+2y'=x\)?
  2. What is the order of \((y')^3+y=0\)?
  3. Which of the following equations is linear?
  4. Which of the following equations is NOT linear?
  5. What is the order of \((y'')^2+y'=x\)?
  6. Classify \(x^2y''+e^{x}y'+y=0\):
  7. Classify \(y''+y\,y'=0\):
  8. For which values of \(n\) is \(y'+y=y^n\) linear?
  9. In a linear equation, the coefficients of \(y,y',\dots\) may depend on:
  10. The pendulum \(\theta''+\dfrac{g}{L}\sin\theta=0\) is an equation that is:

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