Q120 / 160 · AP EAMCET 2022 (05 Jul Shift 2) · Mathematics

By multiplying with $e^{\int P d x}$ on both sides of the equation $\frac{d y}{d x}+P(x) y=Q(x)$, the left side of the equation takes the form $\frac{d}{d x}(y f(x))$, then $f(x)=$

  • A
    $\int y e^{\int P d x} d x$
  • B
    y P(x)
  • Correct answer
    C
    $e^{\int P d x}$
  • D
    $\mathrm{P}(\mathrm{x}) e^{\int P d x}$

Solution

$\frac{d y}{d x}+P(x) y=Q(x)$ multiplying $e^{\int P d x}$ on both sides of eqn. $e^{\int P d x}$ $\frac{d y}{d x}+y$ $e^{\int P d x} P(x)$ $=Q(x) e^{\int P d x}$ equating LHS with $\frac{d}{d x}(y(f(x)))$ $\Rightarrow e^{\int P d x} \frac{d y}{d x}+$ $y e^{\int P d x} P(x)$ $=\frac{d}{d x}(y f(x))$ $\Rightarrow \quad \frac{d}{d x}\left(y e^{\int P d x}\right)$ $=\frac{d}{d x} y f(x)$ $f(x)=e^{\int P d x}$

Hence, the correct answer is option: 'C'.

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