Question

Find the general solution of the given differential equation. x dy dx − y = x2 sin(x) y(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.)

Answer 1

Divide both sides by $x^2$ - note that this means we can't have $x=0$:

$x\dfrac{\mathrm dy}{\mathrm dx}-y=x^2\sin x$

$\dfrac1x\dfrac{\mathrm dy}{\mathrm dx}-\dfrac1{x^2}y=\sin x$

Then the left side reduces to the derivative of a product,

$\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1xy\right]=\sin x$

$\dfrac1xy=\displaystyle\sin x\,\mathrm dx=\cos x+C$

$y=x\cos x+Cx$

This solution is continuous everywhere, but accounting for the singular point $x=0$, the largest interval over which it is defined would be $(0,\infty)$ or $(-\infty,0)$.