Question

Assertion :Statement-1: The value of $y\left(2\right)$ if y satisfies $x^2\frac{dy}{dx}+xy=\sin x,y\left(1\right)=2$ is $\frac{1}{2}{\int }_1^2\frac{\sin t}{t}dt$ Reason: Statement-2: The solution of a linear equation $\frac{dy}{dx}+Py=Q$ can be obtained by multiplying with the factor $e^{{\int }^{Pdx}}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Division by $x^2$ gives
$\frac{dy}{dx}+\frac{1}{x}y=\frac{\sin x}{x^2}$
so the integrating factor is $e^{\int \frac{1}{x}dx}=x$.
$x\frac{dy}{dx}+y=\frac{\sin x}{x}$
$\Rightarrow xy={\int }_0^x\frac{\sin t}{t}dt+C$
Putting x = 1, y = 2, we get $C=2-{\int }_0^1\frac{\sin t}{t}dt$
$y\left(x\right)=\frac{1}{x}{\int }_0^x\frac{\sin t}{t}dt+\frac{C}{x}$
$=\frac{1}{x}{\int }_0^1\frac{\sin t}{t}dt+\frac{2}{x}-\frac{1}{x}{\int }_0^t\frac{\sin t}{t}dt$
$\frac{2}{x}+\frac{1}{x}{\int }_1^x\frac{\sin t}{t}dt$
So $y\left(2\right)=1+\frac{1}{2}{\int }_1^x\frac{\sin t}{t}dt$