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Messages - david

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Chapter 4 / Re: 4.2
« on: November 18, 2019, 09:36:48 AM »
If any of the $a_i$'s are not constant, then we cannot use the method above. Non-constant coefficient differential equations are generally harder to solve. We discussed a few methods in class such as reduction of order or using the Wronskian, but both methods require already knowing one solution.

Chapter 4 / Re: 4.2
« on: November 14, 2019, 04:09:16 PM »
Finding the characteristic equation for higher order equations is very similar to the second order case.

If we have the differential equation $a_ny^{(n)} + a_{n-1}y^{(n-1)} ....  + a_1y' + a_0y = 0$ then the characteristic equation is $a_nr^n + a_{n-1}r^{n-1} .... + a_1r + a_0 = 0$.

To find the roots, we can use the fact that the product of the roots must be  $a_0$ to help guess the roots.

Chapter 3 / Re: Initial conditions evaluated at different $t_0$'s?
« on: October 06, 2019, 07:48:10 PM »
Initial value problems, by definition, are problems where we have a differential equation and specified values of the solution (and its derivatives) at the same point. If we have conditions such as $y(t_0) = y_0$ and $y(t_1) = y_1$ (called boundary conditions), then this problem is called a boundary value problem. To solve these problems, the same process can be used to get the general solution, but you use the boundary conditions instead to find a particular solution. However, unlike initial value problems, where we only needed some continuity conditions for there to be a unique solution, boundary value problems may have infinite, one, or no solutions.

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