Question

Find the general solution of the given differential equation. x2y' + xy = 8 y(x) = give the largest interval over which the general solution is defined. (think about the implications of any singular points. enter your answer using interval notation.) determine whether there are any transient terms in the general solution. (enter the transient terms as a comma-separated list; if there are none, enter none.)

Answer 1

$x^2y'+xy=8$
$xy'+y=\dfrac8x$
$(xy)'=\dfrac8x\implies xy=8\ln|x|+C\implies y=\dfrac{8\ln|x|}x+\dfrac Cx$

which wouldn't be a valid solution over any interval containing $x=0$.