Given the following rst-order IVP: (e x + y) dx + (2 + x + yey ) dy = 0 y (0) = 1 (a) Show that this equation is exact. (b) Solve the exact

Question

Given the following rst-order IVP: (e x + y) dx + (2 + x + yey ) dy = 0 y (0) = 1 (a) Show that this equation is exact. (b) Solve the exact equation for f(x, y) = C. (c) Apply the given initial conditions to nd the value of C that satises the IVP. (d) Check your answer by showing that the given DE above is the dierential of the equation f (x, y) = C (where C is whatever value you got from part (c)), and that the initial condition is satised.

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Melanie 3 weeks 2021-11-18T06:13:08+00:00 1 Answer 0 views 0

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    2021-11-18T06:14:14+00:00

    Answer:

    let M=ex +y and N=2 +x +yey

    (a) σM/σy =1 and σN/σx = 1

    since σM/σy = σN/σx , it is an exact equation

    (b) ∫M dx + ∫terms of N not containing x

    ∫(ex + y) dx +∫yey + 2 dy

    xy + ex +yey -ey +2y=C

    (c) using y(0)=1

    C=3

    (d) from the differential equation given

    by dividing through by dx

    dy/dx = (-y-ex) /(2+x+yey)

    from the solution

    \frac{d}{dx}(xy + ex +yey -ey =2y)=\frac{d}{dx}(3)

    x\frac{dx}{dy} + y + ex + yey\frac{dy}{dx} + ey\frac{dx}{dy} – ey\frac{dy}{dx} + 2\frac{dy}{dx} = 0

    \frac{dy}{dx}= (-y-ex) /(2+x+yey)

    Step-by-step explanation:

    1. integrate with respect to x keeping y constant

    2. integrate terms without x in N

    3. Result of 1 + result 2= C

    4. insert the condition given into 3

    5.  compare the solution of 4 to the differential equation

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