Bernoulli and Euler Differential Equations

• Jan 14th 2009, 07:11 AM
louboutinlover
Bernoulli and Euler Differential Equations

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

Originally Posted by louboutinlover

i am really confused and dont know how to solve the questions on the attachtment.

For 2:

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

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

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

Use partial fractions on the LHS.
• Jan 14th 2009, 08:08 AM
Jester
For the first problem, first put in standard form

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

then divide but $\displaystyle x^4$

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

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

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

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

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

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

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

For a particular solution try a solution of the form

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

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