Question

The slope of the tangent at each point of the curve is equal to the sum of the coordinate of the point. Then, the curve that passes through the origin is

• A. $x+y=e^x-1$
• B. $e^x=x+y$
• C. $y=e^x$
• D. $y=e^x+1$

Answer 1

The correct option is A $x+y=e^x-1$

Given, $\frac{dy}{dx}=x+y\Rightarrow dy=xdx+ydx\Rightarrow dy+dx=xdx+ydx+dx$

So, $d(x+y)=(x+y+1)dx\Rightarrow dx=\frac{d(x+y+1)}{x+y+1}$

Integrating on both sides,

$x=\ln (x+y+1)$

On taking antilog,

$e^x=x+y+1$

Or. $e^x-1=x+y$

So, Option A is correct answer.