Assertion (A) : ' x ' is not an integrating factor for the differ… - Vidyadip

MCQ

Assertion (A) : ' $x$ ' is not an integrating factor for the differential equation $x \frac{d y}{d x}+2 y=e^x$.
Reason (R) : $x\left(x \frac{d y}{d x}+2 y\right)=\frac{d}{d x}\left(x^2 y\right)$.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
✓
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

✓

Answer

Correct option: B.

Both (A) and (R) are true but (R) is not the correct explanation of (A).

(b) : $\frac{d y}{d x}+\frac{2}{x} y=\frac{e^x}{x}$
$
\text { I.F. }=e^{\int \frac{2}{x} d x}=e^{2 \log x}=e^{\log x^2}=x^2
$
$\Rightarrow$ Assertion is correct.
Now, $\frac{d}{d x}\left(x^2 y\right)=x^2 \frac{d y}{d x}+y \cdot 2 x=x\left(x \frac{d y}{d x}+2 y\right)$
$\Rightarrow$ Reason is correct.

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