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Chapter 10 DE2: First Order Separable DEs

In Chapter 9 we discussed the concept of a differential equation and saw how to check if a function is a solution of a given DE. However, except for the case where the DE is of the form

\begin{equation*} y^{(n)}(x)=f(x) \end{equation*}

(where we can find the solutions just by integration) we have not yet seen any algebraic methods for finding the solutions to a DE. (We did see that for first order DEs we can obtain a graphical representation of the solutions via a direction field and a numerical approximation to a particular solution via Euler's method.)

As mentioned in Chapter 9 there is no one general method that will solve all possible DEs. However, for various classes of DEs methods have been found that will find their solution. In this lecture we are going to look at an algebraic method for finding the solutions to the class of DEs called first order separable DEs.