Question

The equation of the curve passing through $(1,3)$ whose slope at any point $(x,y)$ on it is $\frac{y}{x^2}$ is given by

• A. $y={3e}^{-1/x}$
• B. $y={3e}^{1-1/x}$
• C. $y={ce}^{1/x}$
• D. $y={3e}^{1/x}$

Answer 1

The correct option is B $y={3e}^{1-1/x}$

$\frac{dy}{dx}=\frac{y}{x^2}\left(given\right)$

$\int \frac{dy}{dx}=\int \frac{dx}{x^2}$

$\Rightarrow \log y=-\frac{1}{x}+c$

$\Rightarrow y=e^{-\frac{1}{x}+c}$

$\Rightarrow y=e^{-\frac{1}{x}+c}$

$\Rightarrow y=e^{-\frac{1}{x}},c$

since this curve is passing through point $\left(1,3\right)$

so, $3=e^{-\frac{1}{t}}.c$

$\Rightarrow c=3e$

therefore , eqn of the curve is

$y=e^{-\frac{1}{x}}.3e$

$\Rightarrow y=3e^{-\frac{1}{x}+1}$

$\Rightarrow y=3e^{1-\frac{1}{x}}$