3. (6 Points). Solve the initial value problem y’-y.cosx=0, y(pi/2)=2e

Question

3. (6 Points). Solve the initial value problem y’-y.cosx=0, y(pi/2)=2e

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Anna 2 weeks 2021-09-09T12:22:16+00:00 1 Answer 0

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    2021-09-09T12:24:08+00:00

    Answer:

    y=2e^{sin(x)}

    Step-by-step explanation:

    Given equation can be  re written as

    \frac{\mathrm{d} y}{\mathrm{d} x}-ycos(x)=0\\\frac{\mathrm{d} y}{\mathrm{d} x}=ycos(x)\\\\=> \frac{dy}{y}=cox(x)dx\\\\Integrating  \\ \int \frac{dy}{y}=\int cos(x)dx \\\\ln(y)=sin(x)+c…………(i)

    Now it is given that y(π/2) = 2e

    Applying value in (i) we get

    ln(2e) = sin(π/2) + c

    => ln(2) + ln(e) = 1+c

    => ln(2) + 1 = 1 + c

    => c = ln(2)

    Thus equation (i) becomes

    ln(y) = sin(x) + ln(2)

    ln(y) – ln(2) = sin(x)

    ln(y/2) = sin(x)

    y= 2e^{sinx}

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