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Math Help - Verifying by substitution.

  1. #1
    Super Member Showcase_22's Avatar
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    Verifying by substitution.

    Here's the question:

    Verify by substitution that y(t)=4sin(t^2)-3cos(t^2)is a solution to t \frac{d^2y}{dt^2}-\frac{dy}{dt}+4t^3 y=0, t>0.


    I decided to do this question by differentiating y twice (to get \frac{d^2y}{dt^2} and \frac{dy}{dt}) and subsituting these expressions in. Then I would rearrange the equation to make y the subject and prove that what I get is 4sint^2-3cost^2.

    y=4sint^2-3cost^2

    \frac{dy}{dt}=8tcos(t^2)+6tsin(t^2)

    \frac{d^2y}{dt^2}=-16t^2 sin(t^2)+12t^2 cos(t^2)

    After substituting these expressions in I end up with:

    y=4sin(t^2)-\frac{3}{2t^2}sin(t^2)-3cos(t^2)+\frac{2}{t^2}cos (t^2).

    I just wanted to check if my differentiation was correct. I have double and triple checked my substitution and that's fine.

    It just occurred to be that I should have put this in the calculus section. Sorry!
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  2. #2
    Moo
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    Hello,

    Hmmm I don't quite understand what you want to do!

    Your expressions for \frac{dy}{dt} and \frac{d^2y}{dt^2} are correct.

    Now, substitute them in t \frac{d^2y}{dt^2}-\frac{dy}{dt}+4t^3 y and prove it equals 0.
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  3. #3
    Super Member Showcase_22's Avatar
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    That's what I thought as well. Here's how my substitution goes:

    t(-16t^2 sin(t^2)+12t^2 cos (t^2))-(8tcos(t^2)+6tsin(t^2))+4t^3 (4sin (t^2)-3cos(t^2))

    4t^3(-4sin(t^2)+3cos(t^2))-2t(4cos(t^2)+3sin(t^2))+4t^3(4sin(t^2)-3cos(t^2))

    4t^3(-4sin(t^2)+3cos(t^2)+4sin(t^2)-3cos(t^2))-2t(4cos(t^2)+3sin(t^2))

    =-2t(4cos(t^2)+3sin(t^2))=-\frac{dy}{dt}

    I did it a different way to how I did before and I now have this answer. I can't see what i'm doing wrong or if it's a typo on the homework sheet.
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  4. #4
    Moo
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    Quote Originally Posted by Showcase_22 View Post
    That's what I thought as well. Here's how my substitution goes:

    t(-16t^2 sin(t^2)+12t^2 cos (t^2))-(8tcos(t^2)+6tsin(t^2))+4t^3 (4sin (t^2)-3cos(t^2))

    4t^3(-4sin(t^2)+3cos(t^2))-2t(4cos(t^2)+3sin(t^2))+4t^3(4sin(t^2)-3cos(t^2))

    4t^3(-4sin(t^2)+3cos(t^2)+4sin(t^2)-3cos(t^2))-2t(4cos(t^2)+3sin(t^2))

    =-2t(4cos(t^2)+3sin(t^2))=-\frac{dy}{dt}

    I did it a different way to how I did before and I now have this answer. I can't see what i'm doing wrong or if it's a typo on the homework sheet.
    Yeah, the solution would rather be t \frac{d^2y}{dt^2}+4t^3 y=0 but that would be weird because you can simplify by t !

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  5. #5
    Super Member Showcase_22's Avatar
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    Practically every question on the proof by induction homework was a typo. It's very annoying!

    I'll have to make a note of this question to quiz my tutor about. There are two questions I have to do in this way, this is the second but the first one is also a typo!!

    Someone should proof-read these before they are handed out. It's confusing undergraduate students, namely me!!!
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