Hey there! Sign in to join this conversationNew here? Join for free

Can somebody teach me how to do 2nd Order Differential Equations - I'm really curious Watch

    • Thread Starter
    Offline

    0
    ReputationRep:
    I know how to do 1st Order DE, but have never been taught how to do 2nd ODE.

    I wish they were included in the A-level (non-FM) specification.

    Can somebody please write a quick brief, and then set me 3 questions to solve for practice.

    Thanks.
    Offline

    15
    ReputationRep:
    unless you're very good at maths you'll need more than a quick brief. have a search around on google, there are loads of explanations out there
    • Thread Starter
    Offline

    0
    ReputationRep:
    (Original post by didgeridoo12uk)
    unless you're very good at maths you'll need more than a quick brief. have a search around on google, there are loads of explanations out there
    I scored an A* in A-level (year 12) last year? :dontknow:

    We don't do FM A2.

    Offline

    1
    ReputationRep:
    Asking about 2nd order ODEs is a very general topic since it can include very horrible ODEs, some of which cannot even be solved by an analytic solution and have to be approximated using numerical methods. I assume you mean second order linear ODEs? If so, these are pretty easy to do.. just look around on the internet as a poster above suggested
    • Thread Starter
    Offline

    0
    ReputationRep:
    (Original post by Serano)
    Asking about 2nd order ODEs is a very general topic since it can include very horrible ODEs, some of which cannot even be solved by an analytic solution and have to be approximated using numerical methods. I assume you mean second order linear ODEs? If so, these are pretty easy to do.. just look around on the internet as a poster above suggested
    I have the FP2 textbook somewhere in my house - borrowed it off school, shall I just read it? How easy is it to self-teach 2nd ODE?

    Bare in mind, this is just out of pure curiousity, so i'm not very likely to put a lot of effort into it, until I have more free time.
    Offline

    15
    ReputationRep:
    (Original post by SingleMuslimDotCom)
    I know how to do 1st Order DE, but have never been taught how to do 2nd ODE.

    I wish they were included in the A-level (non-FM) specification.

    Can somebody please write a quick brief, and then set me 3 questions to solve for practice.

    Thanks.
    Nobody will be able to explain it to you on here. We have just finished covering them in a lot of detail in my University course and we spent 2 weeks on them. They are not that complicated but there a lot of variations and different types and it takes a while to learn how to identify and solve them all.
    Offline

    11
    ReputationRep:
    It sounds like you're competent enough to manage, so the book should be plenty for you, FP2 ones are pretty simple really, it comes down to just 2 forms if I remember rightly.

    And if there's anything in it you don't quite understand, there's plenty of people always jump at the chance to help with anything on here, not like you just have to leave it. Give it a go, what have you got to lose.

    As for mackem's post, in 2 weeks, you didn't do them in a lot of detail. After finishing a 5 week module almost solely around them, and knowing there's large amounts that weren't covered, if you did them in a lot of detail, then you didn't leave the lecture theatre much!
    There's always lots more to learn.
    Offline

    13
    ReputationRep:
    (Original post by didgeridoo12uk)
    unless you're very good at maths you'll need more than a quick brief. have a search around on google, there are loads of explanations out there
    Tbh, 2nd order homogeneous ODE's (or even most non-homogeneous) seem to me, to be a lot easier to solve than a lot of 1st order ODEs, they follow a much more general rule and it's a bit more mechanical and straightforward than some of the more difficult 1st order ODEs. Things like, finding the integrating factor for a 1st order ODE can be a lot more difficult than solving 2nd orders.
    Offline

    2
    ReputationRep:
    If I remember correctly, the only thing in Further Mathematics is constant coefficient 2nd-order linear ODEs. These have a well-known general solution that can be explained in a few pages - the hardest part is finding particular integrals in the non-homogeneous cases.
    • PS Helper
    Offline

    14
    Here's a very (very) quick briefing on 2nd-order linear homogeneous differential equations with constant coefficients.

    Say you have ay'' +by' + cy = 0. Then when you substitute y=e^{\lambda x}, since y'=\lambda e^{\lambda x} and y''=\lambda^2 e^{\lambda x}, we get:
    e^{\lambda x}(a\lambda^2 + b\lambda + c) = 0

    This is only equal to zero (as is on the RHS) when a\lambda^2 + b\lambda + c = 0 (since e^{\lambda x}>0 for all x).

    This quadratic is called the characteristic equation of the differential equation. Suppose it has two real roots, \alpha and \beta. Then y=e^{\alpha x} and y=e^{\beta x} satisfy the differential equation. But so do multiples of them, and they still do when you add them together, and so the general solution of the differential equation is
    y=Ae^{\alpha x}+Be^{\beta x}
    ...for arbitrary constants A and B (analogous to constants of integration). This solution is known as the "general solution".

    If \alpha = \beta then instead of the above we get (Ax+B)e^{\alpha x}.

    If \alpha,\beta are complex, then we can still write it as y=Ae^{\alpha x}+Be^{\beta x}, or we can use various identities which allow us instead to express it as y=e^{px}(C\cos qx + D\sin qx), where \alpha = p+qi and \beta = p-qi.

    If you were then told that y=0 when x=0 and y'=0 when x=0 then you could find A,B (or C,D) and hence find a particular solution.

    The only way to make any sense of what I've just written is to try some for yourself... reading it through probably won't do anything for you.
    Offline

    0
    ReputationRep:
    (Original post by nuodai)
    Here's a very (very) quick briefing on 2nd-order linear homogeneous differential equations with constant coefficients.

    Say you have ay'' +by' + cy = 0. Then when you substitute y=e^{\lambda x}, since y'=\lambda e^{\lambda x} and y''=\lambda^2 e^{\lambda x}, we get:
    e^{\lambda x}(a\lambda^2 + b\lambda + c) = 0

    This is only equal to zero (as is on the RHS) when a\lambda^2 + b\lambda + c = 0 (since e^{\lambda x}>0 for all x).

    This quadratic is called the characteristic equation of the differential equation. Suppose it has two real roots, \alpha and \beta. Then y=e^{\alpha x} and y=e^{\beta x} satisfy the differential equation. But so do multiples of them, and they still do when you add them together, and so the general solution of the differential equation is
    y=Ae^{\alpha x}+Be^{\beta x}
    ...for arbitrary constants A and B (analogous to constants of integration). This solution is known as the "general solution".

    If \alpha = \beta then instead of the above we get (Ax+B)e^{\alpha x}.

    If \alpha,\beta are complex, then we can still write it as y=Ae^{\alpha x}+Be^{\beta x}, or we can use various identities which allow us instead to express it as y=e^{px}(C\cos qx + D\sin qx), where \alpha = p+qi and \beta = p-qi.

    If you were then told that y=0 when x=0 and y'=0 when x=0 then you could find A,B (or C,D) and hence find a particular solution.

    The only way to make any sense of what I've just written is to try some for yourself... reading it through probably won't do anything for you.
    This is really fantastic mate. I couldn't quite remember 2nd Order homogenous ODEs but this brought it flooding back straight away.

    Good work
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Did TEF Bronze Award affect your UCAS choices?
    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
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • 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

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