Chapter 2 Solvable Equations

Sec 2.1 – 1 st Order Linear Equations  First solvable class of equations  The equation must be able to be expressed in the form  Examples:  There is a “trick” to solving such equations which we will learn in section 2.6  For now we will develop in two cases

Case 1 – f(x) = 0  We call this case homogeneous  Examples:

The homogeneous case can be solved …with straightforward integration, after separating the variables x and y.  Example

Example 2 Read Example on p 33

Works fine with an IC as well  Sec 2.1 #9

In general  Theorem 2.1 on p 33.  If p(x) is continuous for x in (a,b), then the general solution of  Is given by  Where

Proof is pretty easy with Chain Rule  Let  Where P’(x) = p(x). Try to show that y satisfies  What is y’(x)?  You can now do Sec 2.1 p 41 #2-10 even

Case 2 – General (non-homogeneous) case  F(x) can be anything  Examples:

Solved Using Variation Of Parameters  We make substitution according to the form of f(x), and then back substitute at the end  This technique will be useful on many equations as we go through the first part of the course.

We assume the solution looks similar to the homogeneous one, then solve 1. Form the complementary homogeneous equation 2. Find a solution of this, using the technique (or formula) from yesterday. Call it y 1 3. Assume that y, the solution to the original equation, is of the form where u is a function of x, that we can determine

Now substitute into original 1. Remember that so 2. By the product rule, this comes out to 3. We now plug all of these expressions into the original equation to get 4. We now collect the like terms to get

Now where are we? 1. Look at the piece inside the parentheses 2. So our equation reduces to 3. Or, 4. And the solution to the non-homogeneous equation is

Example  Solve