Mathematics

Question

Find the solution of the given initial value problem. y'-y=11te^2t, y(0)=1

1 Answer

  • [tex]y'-y=11te^{2t}[/tex]
    [tex]e^{-t}y'-e^{-t}y=11te^t[/tex]
    [tex]\dfrac{\mathrm d}{\mathrm dt}\left[e^{-t}y\right]=11te^t[/tex]
    [tex]e^{-t}y=11\displaystyle\int te^t\,\mathrm dt[/tex]
    [tex]e^{-t}y=11e^t(t-1)+C[/tex]

    Since [tex]y(0)=1[/tex], you have

    [tex]1=11(-1)+C\implies C=12[/tex]

    and so

    [tex]e^{-t}y=11e^t(t-1)+12[/tex]
    [tex]y=11e^{2t}(t-2)+12e^t[/tex]
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