(Headbang)

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

please help.

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- Jan 14th 2009, 07:11 AMlouboutinloverBernoulli and Euler Differential Equations
(Headbang)

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

please help. - Jan 14th 2009, 07:33 AMMush
- Jan 14th 2009, 08:08 AMJester
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.