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 $y$:

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

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

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

Answer 2

The general solution of a differential equation is to write y as a function of x.

• The general solution of $x \frac{dy}{dx} - y = x^2 \sin(x)$ is $y = -x\cos(x) + cx$.
• The interval of the solution is $(0, \infty)$

Given

$x \frac{dy}{dx} - y = x^2 \sin(x)$

Divide through by x

$\frac{x}{x} \frac{dy}{dx} -\frac{y}{x} = \frac{x^2}{x} \sin(x)$

$\frac{dy}{dx} -\frac{y}{x} = x \sin(x)$

Let P be function of x. Such that:

$P(x) = -\frac 1x$

So, we have:

$\frac{dy}{dx} +yP(x) = x\sin(x)$

Calculate the integrating factor I(x).

So, we have:

$I(x) = e^{\int P(x) dx$

Substitute $P(x) = -\frac 1x$

$I(x) = e^{\int-\frac 1x dx$

Rewrite as:

$I(x) = e^{-\int\frac 1x dx$

Integrate

$I(x) = e^{-\ln(x)$

$I(x) = \frac 1x$

So, we have:

$\frac{dy}{dx} -\frac{y}{x} = x \sin(x)$

$[\frac{dy}{dx} -\frac{y}{x}] \frac 1x = [x \sin(x)] \frac 1x$

$[\frac{dy}{dx} -\frac{y}{x}] \frac 1x =\sin(x)$

Introduce $I(x) = \frac 1x$.

So, we have:

$\frac{d}{dx}(\frac yx) = \sin(x)$

Multiply both sides by dx

$d(\frac yx) = \sin(x)\ dx$

Integrate with respect to x

$\frac yx = -\cos(x) + c$

Multiply through by x

$y = -x\cos(x) + cx$

So, the general solution is: $y = -x\cos(x) + cx$, and the interval is $(0, \infty)$

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