You CAN Ace Differential Equations
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Single Variable Calculus |
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Multi-Variable Calculus |
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Acceleration Vector |
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Arc Length Function |
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Conservative Vector Fields |
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Cylindrical Coordinates |
Lagrange Multipliers |
Line Integrals |
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Path Integrals |
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Principal Unit Normal Vector |
Differential Equations |
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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 |
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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 |
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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.
video by MIT OCW
Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems |
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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.