Bernoulli and Euler Differential Equations

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January 14th 2009, 07:11 AM
louboutinlover

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Bernoulli and Euler Differential Equations

(Headbang)

i am really confused and dont know how to solve the questions on the attachtment.
please help.
January 14th 2009, 07:33 AM
Mush

Quote:

Originally Posted by louboutinlover

(Headbang)

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.
January 14th 2009, 08:08 AM
Danny

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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