2nd Order In-Homogeneous Particular Solution?

Thread starter raaznar
Start date May 26, 2013
Tags

2nd order Particular solution

Click For Summary

SUMMARY

The discussion focuses on finding the second-order in-homogeneous particular solution for differential equations. The characteristic equation is identified as r² + 4r + 3 = 0, yielding roots r = -1 and r = -3. The general solution for the associated homogeneous equation is Ae^{-3x} + Be^{-x}. For the in-homogeneous terms, the suggested forms include e^{2x}(Ccos(x) + Dsin(x)) for 2e^{2x}cos(x) and e^{-4x}(Ex + F) for 3xe^{-4x}, with G proposed for the constant term 3.

PREREQUISITES

Understanding of differential equations and their solutions
Familiarity with characteristic equations and their roots
Knowledge of in-homogeneous solutions and particular solutions
Basic skills in manipulating exponential and trigonometric functions

NEXT STEPS

Study the method of undetermined coefficients for solving in-homogeneous differential equations
Learn about the Laplace transform and its applications in solving differential equations
Explore the variation of parameters technique for finding particular solutions
Investigate the role of exponential functions in differential equations

USEFUL FOR

Students and professionals in mathematics, engineering, and physics who are working with differential equations and seeking to enhance their problem-solving skills in this area.

May 26, 2013

raaznar

Did it :)

Last edited: May 26, 2013

Physics news on Phys.org

May 26, 2013

HallsofIvy

Science Advisor

Homework Helper

Yes, the characteristic equation is [itex]r^2+ 4r+ 3= (r+ 3)(r+ 1)= 0[/itex] which has roots r= -1 and r= -3 so the general solution to the associated homogeneous equation is [itex]Ae^{-3x}+ Be^{-x}[/itex].

For [itex]2e^{2x}cos(x)[/itex] try [itex]e^{2x}(Ccos(x)+ Dsin(x)[/itex]. For ##3xe^{-4x}##, try [itex]e^{-4x}(Ex+ F)[/itex]. For 3, try G.

Last edited by a moderator: May 26, 2013