First Order Linear DEs   Solving 1st Order Linear Homogeneous DEs       Finding a Particular Solution  Examples     Conclusion
                                                       MATH312
                       Section 2.3: First Order Linear Equations
                                               Prof. Jonathan Duncan
                                                    Walla Walla University
                                                 Spring Quarter, 2008
    First Order Linear DEs   Solving 1st Order Linear Homogeneous DEs       Finding a Particular Solution  Examples     Conclusion
    Outline
           1    First Order Linear DEs
           2    Solving 1st Order Linear Homogeneous DEs
           3    Finding a Particular Solution
           4    Examples
           5    Conclusion
    First Order Linear DEs   Solving 1st Order Linear Homogeneous DEs       Finding a Particular Solution  Examples     Conclusion
    Not Everything Simple is Separable
           While separable differential equations are relatively easy to solve,
           unfortunately, many simple differential equations are not separable.
           Example
           The following simple differential equations are not separable.
                   dy =x +y
                   dx
                   y′ = e3x −y
           There are several questions we need to ask about these DEs.
                   What is it about these DEs which makes them “simple?”
                   Can we find a solution process for these DEs?
    First Order Linear DEs   Solving 1st Order Linear Homogeneous DEs       Finding a Particular Solution  Examples     Conclusion
    Definitions
           Westart by answering the first question.
           Linear 1st Order DE
           Afirst order differential equation of the form
           a (x)dy +a (x)y = g(x) is a linear equation.
             1      dx          0
           Homogeneous DE
           When g(x) = 0 in the above equation, we call it a homogeneous
           equation. Otherwise, it is non-homogeneous.
           Standard Form
           As we will see, our method of solution requires that we put each
           linear DE into standard form: dy + P(x)y = f(x)
                                                              dx