Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    2
    ReputationRep:
    Let L be defined by L(y)=y\prime\prime+ay\prime+by for a,b \in \mathbb{R} constants.

    Let R be a complex valued function so that R(x)=P(x)+iQ(x) for functions P(x) and Q(x). Prove the function f(x)=u(x)+iv(x) satisfies the differential equation L(y)=R(x) \enspace x \in I if and only if u and v satisfy the differential equations L(u)=P(x) \enspace L(v)=Q(x) \enspace x \in I.

    I literally don't even know where to start, are there some formula or trick that can help?
    Offline

    3
    ReputationRep:
    (Original post by AishaGirl)
    Let R be a complex valued function so that R(x)+iQ(x) for functions P(x) and Q(x).
    I may be wrong or misreading something here, but should R(x)+iQ(x) read R(x) = P(x)+iQ(x)?
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by crashMATHS)
    I may be wrong or misreading something here, but should R(x)+iQ(x) read R(x) = P(x)+iQ(x)?
    Yes you are right, sorry it was a typo I fixed it now.
    Offline

    11
    ReputationRep:
    (Original post by AishaGirl)
    Let L be defined by L(y)=y\prime\prime+ay\prime+by for a,b \in \mathbb{R} constants.

    Let R be a complex valued function so that R(x)=P(x)+iQ(x) for functions P(x) and Q(x). Prove the function f(x)=u(x)+iv(x) satisfies the differential equation L(y)=R(x) \enspace x \in I if and only if u and v satisfy the differential equations L(u)=P(x) \enspace L(v)=Q(x) \enspace x \in I.

    I literally don't even know where to start, are there some formula or trick that can help?
    Hint: a+ib=c+id if and only if a=c,b=d
    • Thread Starter
    Offline

    2
    ReputationRep:
    ok so I think I have an idea how to start.

    If I assume that y=f(x)=u(x)+iv(x) satisfies the differential equation then can I say something like

    L(y)=R(x) \implies f\prime\prime(x)+af\prime(x)+bf(  x)=P(x)+iQ(x)

    \implies u\prime\prime(x)+iv\prime\prime(  x)+au \prime(x)+iav\prime(x)+bu\prime(  x)+bv\prime(x)=P(x)+iQ(x)

    Is the correct way to go?
    Offline

    11
    ReputationRep:
    (Original post by AishaGirl)
    Is the correct way to go?
    Yes. Now collect real and imaginary parts.
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by atsruser)
    Yes. Now collect real and imaginary parts.
    Like this?

    u\prime\prime(x)+au\prime(x)+bu(  x)=P(x)

    v\prime\prime(x)+av\prime(x)+bv(  x)=Q(x)

    So L(u)=P(x) and L(v)=Q(x) ?

    Looks good so far?
    • Thread Starter
    Offline

    2
    ReputationRep:
    I think I have proved it now.

    Assume L(u)=P(x) and L(v)=Q(x) for all x\in I then

    u\prime\prime(x)+au \prime(x)+bu(x)+i(v\prime\prime(  x)+av \prime(x)+bv(x))=P(x)+iQ(x)

     \implies f\prime\prime(x)+af\prime(x)+bf(  x)=R(x) \implies L(y)=R(x) for all x\in I

    Is this correct? It's quite the mouthful

    atsruser alow RDKGames DFranklin can someone tell me if it's correct?
    Offline

    3
    ReputationRep:
    (Original post by AishaGirl)
    I think I have proved it now.

    Assume L(u)=P(x) and L(v)=Q(x) for all x\in I then

    u\prime\prime(x)+au \prime(x)+bu(x)+i(v\prime\prime(  x)+av \prime(x)+bv(x))=P(x)+iQ(x)

     \implies f\prime\prime(x)+af\prime(x)+bf(  x)=R(x) \implies L(y)=R(x) for all x\in I

    Is this correct? It's quite the mouthful

    atsruser alow RDKGames DFranklin can someone tell me if it's correct?
    Looks good
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by crashMATHS)
    Looks good
    Cheers!
    Offline

    11
    ReputationRep:
    (Original post by AishaGirl)
    I think I have proved it now.

    Assume L(u)=P(x) and L(v)=Q(x) for all x\in I then

    u\prime\prime(x)+au \prime(x)+bu(x)+i(v\prime\prime(  x)+av \prime(x)+bv(x))=P(x)+iQ(x)

     \implies f\prime\prime(x)+af\prime(x)+bf(  x)=R(x) \implies L(y)=R(x) for all x\in I
    You're halfway there. The question is an "if and only if" proof. Suppose A and B are statements, and A is true if and only if B is true - that means that A implies B and that B implies A i.e.

    A \Rightarrow B and B \Rightarrow A or equivalently A \Leftrightarrow B

    You have proved the "if" part i.e. A \Rightarrow B part. You still need to prove the "only if" part i.e. B \Rightarrow A. To do so, you need to show that if f(x) satisfies the given DE, then L(u) = P(x), L(v)=Q(x)
    Offline

    3
    ReputationRep:
    (Original post by atsruser)
    You're halfway there. The question is an "if and only if" proof. Suppose A and B are statements, and A is true if and only if B is true - that means that A implies B and that B implies A i.e.

    A \Rightarrow B and B \Rightarrow A or equivalently A \Leftrightarrow B

    You have proved the "if" part i.e. A \Rightarrow B part. You still need to prove the "only if" part i.e. B \Rightarrow A. To do so, you need to show that if f(x) satisfies the given DE, then L(u) = P(x), L(v)=Q(x)
    I believe she's done both directions on two separate posts
 
 
 
  • 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.