Question

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

Answer 1

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

Divide both sides by $x^2$. In doing so, we force any possible solutions to exist on either $(-\infty,0)$ or $\boxed{(0,\infty)}$ (the "positive" interval in such a situation is usually taken over the "negative" one) because $x$ cannot be 0 in order for us to do this.

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

Condense the left side as the derivative of a product, then integrate both sides and solve for $y$:

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

$\dfrac yx=\displaystyle\int\sin x\,\mathrm dx$

$\boxed{y=Cx-x\cos x}$