Turn on thread page Beta
    Offline

    11
    ReputationRep:
    You have already obtained the general solution.

    (Original post by Scary)
    the first part of the question was to use the auxiliary equation to find the yc and then finding the particular integral, the equation was d^2y/dx^2 -dy/dx -6y = e^-2x, the general solution i got was -1/5xe^(-2x) +Ae^(3x)+Be^(-2x)
    Offline

    11
    ReputationRep:
    How did you get on?
    • Thread Starter
    Offline

    13
    ReputationRep:
    thanks for all the help, one of the constants isnt the same though, its divided by 5 is it possible to let another constant say k=c/5?
    Offline

    11
    ReputationRep:
    Hi, yes,  \frac{c}{5} is still a constant.

    This question overall demonstrates another way to solve second ODEs without having to try various functions to be the particular integral.

    Also, if the particular integral (PI) is the same as one of the solutions to the complementary function (in this case  PI(x) = Ae^{-2x} ), then when you substitute the PI back into the second order ODE, you end up with your PI equal to zero.
    • Thread Starter
    Offline

    13
    ReputationRep:
    Hi yes to avoid the pi being a failing case you would change the trial function slightly? Or is there some other way around it?
    Offline

    11
    ReputationRep:
    (Original post by Scary)
    Hi yes to avoid the pi being a failing case you would change the trial function slightly? Or is there some other way around it?
    One method is to multiply the particular integral (PI) with ascending powers of  x , staring from  x .

    In this case, instead of using the PI as  Ae^{-2x} we can try  Axe^{-2x} (where  A is a real value).

    Then substitute in  \frac{dy}{dx} \text{ and } \frac{d^{2}y}{dx^{2}} into the original ODE as normal.

    If this still provides a formula equal to zero, raise to the next power of  x for your new PI.

    Other methods to solve second order ODEs are Laplace transforms, Wronskians, variation of parameters, Frobenius method (to name a few).
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: August 11, 2016
Poll
Do you think parents should charge rent?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.