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Math Help - Bernoulli and Euler Differential Equations

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    Bernoulli and Euler Differential Equations



    i am really confused and dont know how to solve the questions on the attachtment.
    please help.
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  2. #2
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    Quote Originally Posted by louboutinlover View Post


    i am really confused and dont know how to solve the questions on the attachtment.
    please help.
    For 2:

     \frac{dN}{dt} = -(N^2+2N)

     \frac{dN}{N^2+2N} = -dt

     \frac{dN}{N(N+2)} = -dt

    Use partial fractions on the LHS.
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  3. #3
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    For the first problem, first put in standard form

     \frac{dx}{dt} - \frac{x}{t} = - \frac{1}{3} x^4 t^2

    then divide but x^4

     \frac{1}{x^4}\,\frac{dx}{dt} - \frac{1}{t} \, \frac{1}{x^3} = - \frac{1}{3} t^3

    Let u = \frac{1}{x^3} which transform the equation to

    \frac{du}{dt} + \frac{3 u}{t} = t^2.

    Now this is linear so you should be able to solve.

    For the third, if we seek solutions of the form
    x = t^m then characteristic equation is

    m(m-1) + 7m + 9 = 0 from which we obtain
    m = -3,\;-3 which give the complementary solution is

    x = c_1 t^{-3} + c_2 t^{-3} \ln t

    For a particular solution try a solution of the form

    y = A \cos \ln t + B \sin \ln t

    substitute in the ODE and compare like terms - this will give two equations for A and B.
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