MATH 312 Section 4.3: Homogeneous Linear Equations with Constant Coefficients Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007
Outline 1 Getting Started 2 Second Order Equations Two Real Roots Single Real Root Two Complex Solutions 3 Higher Order Equations 4 Conclusion
Starting from Scratch In our search for a solution procedure, we have yet to find a process for finding a solution from scratch. Solution Procedure In this section, we study methods for solving a homogeneous linear differential equation with constant coefficients. That is: Recall: d n y a n dx n + a d n 1 y n 1 dx n 1 + + a dy 1 dx + a 0y = 0 An equation of the form y + ay = 0 can be solved using separation of variables. The solution is y = Ce ax. Can we generalize this to higher order equations?
Auxiliary Equations To see if such a solution would work, we will consider a general solution of this form for a second degree homogeneous equation. Example Determine for what values of m the function y = e mx would be a solution to ay + by + cy = 0. y = e mx y = me mx y = m 2 e mx e mx (am 2 + bm + c) = 0 am 2 + bm + c = 0 (auxiliary equation)
Two Real Roots An Example The auxiliary equation we found above can be solved to find the general solution. However, there are several different types of solutions to such an equation. Two Real Roots The auxiliary equation am 2 + bm + c = 0 may have two distinct real roots, x = m 1 and x = m 2. Example Solve the differential equation y 3y + 2y = 0.
Two Real Roots General Solutions In general, what happens when we have two distinct real roots? General Solution - Two Real Roots A differential equation of the form ay + by + cy = 0 in which the auxiliary equation am 2 + bm + c = 0 has two distinct real roots m 1 and m 2 has a general solution which is valid on (, ). y = C 1 e m 1x + C 2 e m 2x There are two things we should check: e m 1x and e m 2x are both solutions e m 1x and e m 2x are linearly independent
Single Real Root An Example Not all polynomials have distinct roots. We may have roots with multiplicity. A Single Real Roots The auxiliary equation am 2 + bm + c = 0 may have a single real roots, x = m 1 with multiplicity two. Example Solve the differential equation y 10y + 25y = 0.
Single Real Root General Solutions In general, what happens when we have a single real root with multiplicity? General Solution - Two Real Roots A differential equation of the form ay + by + cy = 0 in which the auxiliary equation am 2 + bm + c = 0 has a single real root m 1 of multiplicity two has a general solution which is valid on (, ). y = C 1 e m 1x + C 2 xe m 1x There are two steps to this process: Verify that e m 1x is a solution Use reduction of order to get our second solution.
Two Complex Solutions An Example The final option for a quadratic auxiliary equation is a pair of complex roots. A Single Real Roots The auxiliary equation am 2 + bm + c = 0 may have two complex roots of the form α + βi and α βi. Example Solve the differential equation 2y 3y + 4y = 0. Euler s Formula Recall that Euler s Formal states that e βi = cos β + i sin β
Two Complex Solutions General Solutions What happens when we have a conjugate pair of complex roots? General Solution - Two Real Roots A differential equation of the form ay + by + cy = 0 in which the auxiliary equation am 2 + bm + c = 0 has complex roots α + βi and α βi has the general solution which is valid on (, ). y = C 1 e αx cos βx + C 2 e αx sin βx There are several steps to this process: Verify that e α±βi are solutions. Use Euler s formula to rewrite as e αx (cos βx ± i sin βx). Use superpositioning to get the final solution.
General Solution Procedure These processes will work on higher order linear homogeneous equations with constant coefficients as well. Distinct Real Roots Let R(m) = a n m n + a n 1 m n 1 + + a 1 m + a 0 = 0 be the auxiliary equation for the differential equation d n y a n dx n + a d n 1 y n 1 dx n 1 + + a dy 1 dx + a 0y = 0 If R(m) has n distinct real roots m 1, m 2,..., m n, then the general solution to the differential equation is: What about Other Cases? y = C 1 e m1x + C 2 e m2x + + C n e mnx What do we do with multiple (now 2) and/or complex roots?
Roots with Multiplicity We deal with multiplicity as follows. Multiplicity and Real Roots If m 1 is a real root of multiplicity k for the auxiliary equation R(m) = 0, then the following are k linearly independent solutions of the differential equation. e m1x, xe m1x, x 2 e m1x,..., x k 1 e m1x Multiplicity and Complex Roots If α + βi is a complex solution of multiplicity k for the auxiliary equation R(m) = 0, then the following are 2k linearly independent solutions of the differential equation. e αx cos βx, xe αx cos βx,..., x k 1 e αx cos βx e αx sin βx, xe αx sin βx,..., x k 1 e αx sin βx
An Example We finish this section with an example of a higher order differential equation and IVP Example Solve the initial value problem y y y + y = 0 subject to y(0) = 0, y (0) = 7, and y (0) = 2. m 3 m 2 m + 1 = 0 m 2 (m 1) (m 1) = 0 m 1 = 1, m 2 = 1, m 3 = 1 y = C 1 e x + C 2 xe x + C 3 e x
Important Concepts Things to Remember from Section 4.3 1 Finding the auxiliary equation 2 Solutions for 2nd order DEs where the auxiliary equation has Two distinct real roots A single real root with multiplicity Two complex roots 3 Using auxiliary equations to solve higher order DEs