Find the general solution of the given differential equation. x dy dx − y = x2 sin(x) y(x) = cx−xcos(x) correct: your answer is correct. 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.) incorrect: your answer is incorrect. 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.)

Note that in order to do this division, we cannot allow [tex]x=0[/tex]. This means the largest interval on which a solution can exist is either [tex](0,\infty)[/tex] or [tex](-\infty,0)[/tex].

If [tex]y(x)[/tex] is a solution to the ODE, then any term that vanishes as [tex]x\to\infty[/tex] (or [tex]-\infty[/tex], depending on which interval above is used) is a transient term.

Solve the ODE:

[tex]\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac yx\right]=\sin x\implies\dfrac yx=C-\cos x\implies y=Cx-x\cos x[/tex]

As [tex]x\to\infty[/tex], [tex]\cos x[/tex] will oscillate between -1 and 1, so [tex]x\cos x[/tex] will oscillate between [tex]-\infty[/tex] and [tex]\infty[/tex], so the limit of [tex]y(x)[/tex] as [tex]x\to\infty[/tex] does not exists. There are no transient terms.