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