You CAN Ace Differential Equations

Topics You Need To Understand For This Page

Related Topics and Links

external links you may find helpful

Wikipedia - Wronskian

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

free ideas to save on books

The Wronskian is a determinant that is used to show linear independence of a set of solutions to a differential equation. Although our discussion is limited to two dimensions, the concepts can be easily extended to higher dimensions.

Theorem: Existence and the Wronskian

For the second order linear homogeneous differential equation \( y''+p(t)y' + q(t)y = 0\) with initial conditions \( y(t_0) = a_0, y'(t_0) = a_1\), suppose that we have found two solutions \(y_1(t)\) and \(y_2(t)\). Then it is possible to determine the constants \( c_1, c_2 \) so that \( y(t) = c_1 y_1(t) + c_2 y_2(t)\) satisfies the differential equation and initial conditions if and only if the Wronskian \[ W = \begin{vmatrix} y_1(t_0) & y_2(t_0) \\ y_1'(t_0) & y_2'(t_0) \end{vmatrix} \] is not zero.

To use this theorem, we don't always need initial conditions. If we calculate the Wronskian \[ W = \begin{vmatrix} y_1(t) & y_2(t) \\ y_1'(t) & y_2'(t) \end{vmatrix} \] and determine that this Wronskian is non-zero everywhere (i.e. for all values of \(t\)), then we can still apply the theorem and say that we can construct solutions together with initial conditions specified at any value of t.

Determining Constants

We can use the Wronskian \(W\) to actually calculate the constants \( c_1, c_2 \). By substituting the initial conditions, we get the two equations with two unknowns \[ \begin{array}{rcrcr} c_1 y_1(t_0) & + & c_2 y_2(t_0) & = & a_0 \\ c_1 y_1'(t_0) & + & c_2 y_2'(t_0) & = & a_1 \end{array} \]

We can use Cramer's Rule to solve a system of linear equations. For a reminder on how to do this, check out the linear algebra page.

Here is a video discussing the Wronskian, superposition and uniqueness.

MIT OCW - Theory of General Second-order Linear Homogeneous ODE's [50mins-31secs]

video by MIT OCW


Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.


Instructions - - Unless otherwise instructed, solve these problems giving your answers in exact terms, completely factored.

Real Time Web Analytics