Q52 / 155 · WBJEE 2024 · Mathematics

If $x y^{\prime}+y-e^x=0, y(a)=b$, then $\lim _{x \rightarrow 1} y(x)$ is

  • A
    $e+2 a b-e^a$
  • B
    $e^2+a b-e^{-a}$
  • C
    $e-a b+e^a$
  • Correct answer
    D
    $e+a b-e^a,\left(y^{\prime}=\frac{d y}{d x}\right)$

Solution

$\begin{aligned} & \text {Hint : } \quad xy^{\prime}+y-e^x=0 \\ & \Rightarrow \quad x \frac{d y}{d x}+y=e^x \\ & \Rightarrow \quad x d y+y d x=e^x d x \\ & \Rightarrow \quad \int d(x y)=\int e^x d x \\ & \Rightarrow \quad x y=e^x+c \\ & \quad y(a)=b \Rightarrow c=a b-e^a \\ & \therefore \quad y(x)=\frac{e^x+a b-e^a}{x} \\ & \therefore \quad \lim _{x \rightarrow 1} y(x)=e+a b-e^a\end{aligned}$

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

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