My Math Forum High order non-homog odes, general question

 Calculus Calculus Math Forum

 May 4th, 2012, 06:11 AM #1 Newbie   Joined: Mar 2012 Posts: 12 Thanks: 0 High order non-homog odes, general question Hey, I have a general question regarding the particular solution to high order odes, for example of you have an ode y'''' + y''' = x - 2e^(2x) a particular solution of x^3(Ax + B) + C e^(2x) works, But I only knew that from pretty much trial and error, and guessing. Is there a general method of choosing particular solutions for higher order odes? Thanks, Linda
 May 4th, 2012, 07:25 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: High order non-homog odes, general question The auxiliary equation associated with the complementary equation has the root r = 0 of multiplicity 3 and the root r = -1 meaning the complementary solution is: $y_h(x)=c_1+c_2x+c_3x^2+c_4e^{-x}$ Now, looking at the right side of the ODE, we may assume a particular solution corresponding to the first term of the form: $y_{p_1}(x)=x^s$$Ax+B$$$ And for the second term: $y_{p_2}(x)=x^s$$Ce^{2x}$$$ Now, for both solutions, the non-negative integer s is chosen to be the smallest integer so that no term in the particular solution is a solution to the corresponding homogeneous equation. Hence, we have: $y_{p_1}(x)=x^3$$Ax+B$$$ $y_{p_2}(x)=x^0$$Ce^{2x}$$=Ce^{2x}$ And so, by superposition, we have: $y_p(x)=y_{p_1}(x)+y_{p_2}(x)=x^3$$Ax+B$$+Ce^{2x}$
 May 6th, 2012, 08:46 PM #3 Newbie   Joined: Mar 2012 Posts: 12 Thanks: 0 Re: High order non-homog odes, general question Sorry I made a small error in my post, x^3(Ax + B) + Cxe^(2x) works, I forgot the x next to the exponential term, but now for y(x) = x^s(Ce^2x), so s=1 can work as well? but it's just easier to pick the lowest right so long as it doesn't match the solution to the homogeneous equation?
 May 6th, 2012, 09:21 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: High order non-homog odes, general question Does the exponential term on the right of the equal sign have an x next to it in the ODE? What is the original ODE?

 Tags general, high, nonhomog, odes, order, question

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post mirror Applied Math 2 September 8th, 2013 08:04 AM Raghav C Marthi New Users 1 April 19th, 2013 02:46 AM r-soy Calculus 15 March 1st, 2013 09:39 AM Linda2 Calculus 3 March 31st, 2012 06:42 PM lanvin12 Algebra 2 November 27th, 2008 05:47 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top