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Math Help - initial value problems

  1. #1
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    initial value problems

    solve the following initial value problems:

    dy/dt cos t + y sin t = 1 where y(Pi/4) = sqrt(2)

    (1 + x^2) dy/dx = y where y(0) = 1

    t* dy/dt - y = t^3 sin t where y(Pi) = 0
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by razorfever View Post
    solve the following initial value problems:

    dy/dt cos t + y sin t = 1 where y(Pi/4) = sqrt(2)
    Divide through by \cos t to get \frac{\,dy}{\,dt}+y\tan t=\sec t

    The integrating factor is \varrho\!\left(t\right)=\sec t (Verify)

    Thus, the DE becomes \frac{\,d}{\,dt}\left[y\sec t\right]=\sec^2 t

    Can you take it from here??

    (1 + x^2) dy/dx = y where y(0) = 1
    Separating the variables, you have \frac{\,dy}{y}=\frac{\,dx}{1+x^2}

    Can you take it from here?

    t* dy/dt - y = t^3 sin t where y(Pi) = 0
    Divide through by t to get \frac{\,dy}{\,dt}-\frac{1}{t}y=t^2\sin t

    The integrating factor is \varrho\!\left(t\right)=\frac{1}{t}

    Thus, the DE becomes \frac{\,d}{\,dt}\left[y\frac{1}{t}\right]=t\sin t

    Can you take it from here?

    Does this make sense?
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