Question

If \(y+\frac{\mathrm{d}}{\mathrm{d} x}(x y)=x(\sin x+\log x)\) then

• A. \(y=\mathrm{cos}x+\frac{2\mathrm{sin}x}{x}+\frac{2}{x^{2}}\mathrm{cos}x+\frac{x}{3}\mathrm{log}x-\frac{x}{9}+\frac{\mathrm{c}}{x^{2}}\), where c is the constant of integration.
• B. \(y=-\mathrm{cos}x-\frac{2}{x}\mathrm{sin}x+\frac{2}{x^{2}}\mathrm{cos}x+\frac{x}{3}\mathrm{log}x-\frac{x}{9}+\frac{\mathrm{c}}{x^{2}}\) where c is the constant of integration.
• C. \(\begin{array}{cc} y=-\mathrm{cos}x+\frac{2}{x}\mathrm{sin}x+ & \frac{2}{x^{2}}\mathrm{cos}x+\frac{x}{3}\mathrm{log}x-\frac{x}{9}+\frac{c}{x^{2}}\text{ where }\mathrm{c}\text{ is the constant of integration. } \end{array}\)
• D. \(y=\mathrm{cos}x-\frac{2}{x}\mathrm{sin}x+\frac{2}{x^{3}}\mathrm{cos}x+\frac{x}{3}\mathrm{log}x-\frac{x}{9}+\frac{\mathrm{c}}{x^{2}}\) where \(c\) is the constant of integration.

Hint:

Expand d(xy)/dx to get a linear equation and use the integrating factor x^2.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Expand \(\dfrac{d}{dx}(xy)=y+x\dfrac{dy}{dx}\) and reduce the equation to the linear form \(\dfrac{dy}{dx}+Py=Q\).

Step 2: Linear form
\(x\dfrac{dy}{dx}+2y=x(\sin x+\log x)\Rightarrow\dfrac{dy}{dx}+\dfrac{2}{x}y=\sin x+\log x\).

Step 3: Integrating factor
\(\text{I.F.}=e^{\int\frac{2}{x}dx}=x^{2}\), so \(x^{2}y=\int(x^{2}\sin x+x^{2}\log x)dx\).

Step 4: Integrate by parts
\(\int x^{2}\sin x\,dx=-x^{2}\cos x+2x\sin x+2\cos x\) and \(\int x^{2}\log x\,dx=\dfrac{x^{3}}{3}\log x-\dfrac{x^{3}}{9}\).

Step 5: Divide by x^2
\(y=-\cos x+\dfrac{2}{x}\sin x+\dfrac{2}{x^{2}}\cos x+\dfrac{x}{3}\log x-\dfrac{x}{9}+\dfrac{c}{x^{2}}\).

Final Answer:
The solution is y = -cos x + (2/x) sin x + (2/x^2) cos x + (x/3) log x - x/9 + c/x^2. \[ \boxed{\text{(C) }y=-\cos x+\dfrac{2}{x}\sin x+\dfrac{2}{x^{2}}\cos x+\dfrac{x}{3}\log x-\dfrac{x}{9}+\dfrac{c}{x^{2}}} \]