Higher-order linear equations · Definition

Wronskian

differential equationshigher order

Before you start you need to know

It is the test that decides whether two solutions are enough to describe all the solutions of a second-order linear homogeneous equation.

Definition

The Wronskian \(W\) of two differentiable functions \(y_1\) and \(y_2\) is the determinant of the matrix formed by them and their first derivatives.

\[W(y_1,y_2)=\begin{vmatrix}y_1&y_2\\y_1'&y_2'\end{vmatrix}=y_1y_2'-y_1'y_2\]

Two functions are linearly dependent when one is a multiple of the other. The Wronskian detects this with derivatives: if \(y_2=ky_1\), the two columns of the matrix are proportional and the determinant is \(0\) at every point. For example, \(W(e^t,2e^t)=0\), while \(W(\cos t,\sin t)=1\).

For arbitrary functions the converse fails: \(t^2\) and \(t|t|\) have Wronskian zero everywhere and are not proportional. Things change when \(y_1,y_2\) solve \(y''+py'+qy=0\) with \(p,q\) continuous. Abel's formula states that \(W(t)=W(t_0)\,e^{-\int_{t_0}^{t}p(s)\,ds}\), an exponential that never vanishes; so \(W\) is zero at every point or at none.

In that case, if \(W\neq0\) at some point, \(y_1,y_2\) form a fundamental set: any initial condition \(y(t_0),y'(t_0)\) can be matched with \(c_1y_1+c_2y_2\), and the general solution is exactly that combination. With \(e^{2t}\) and \(e^{3t}\), \(W=e^{5t}\neq0\), so they are a basis of solutions of \(y''-5y'+6y=0\). The next step is to find such solutions with the characteristic equation.

Think about it

Can the Wronskian of two solutions of \(y''+py'+qy=0\) be \(0\) at \(t=1\) and \(5\) at \(t=2\)?

No. By Abel's formula, \(W\) is a multiple of an exponential that never vanishes, so if it is zero at one point it is zero everywhere.

Exercises 10

  1. Compute the Wronskian \(W(e^{2t},e^{3t})\).
  2. Compute \(W(t,t^2)\).
  3. Compute \(W(\cos t,\sin t)\).
  4. Compute \(W(e^{t},2e^{t})\) and interpret the result.
  5. Compute \(W(e^{-t},te^{-t})\).
  6. Compute \(W(\sin 2t,\cos 2t)\).
  7. For \(t>0\), compute \(W(t,\,t\ln t)\).
  8. Let \(y_1,y_2\) be solutions of \(y''+py'+qy=0\), with \(p,q\) continuous, and suppose \(W(y_1,y_2)(t_0)\neq0\) at a point \(t_0\). What follows?
  9. Let \(y_1,y_2\) be solutions of \(y''+py'+qy=0\) with \(p,q\) continuous, and \(W(y_1,y_2)(t_0)=0\) at some \(t_0\). Which is true?
  10. Two solutions of \(y''+3y'+2y=0\) have \(W(0)=2\). What is \(W(t)\)?

References

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §3.2
  • Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §4.1