Published on: Tue Feb 01 2011
Differential equations are equations involving derivatives. For example:
y''+xy'+2y = sin x
This is a second order, linear differential equation.
The order is the highest derivative, in the above case, it's 2 because y''.
Linear can be thought of as a 1st degree polynomial.
Differential equations involving more than 1 independent variable are called partial differential equations. This course is mainly concerned with ordinary differential equations.
We learned two methods of solving ODQ's today. The first method was Separation of Variables. This means rearranging the equation so that one variable is on one side and the other variable is on the other side, then integrating and finally solving for the function variable.
The second method was not given a clear name, but I think it could be called using an integration factor. We used this method because the variables could not be easily separated, but we observed the structure of the eq was similar to the output of the product rule for derivatives. So we added an extra term, mu, to both sides and integrated using it.
Finally we learned the Theorem of 1st Order Linear Differential Equations. Which states y'+p(x)y=q(x) If p,q are continuous on a < x < b, containing x, there is a unique solution satisfying the equation, and an initial condition y(x) = y.
An initial condition is ie y(0)=1, it is used for setting the constant.