Question

The equation of a curve such that the sum of the co-ordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by $5$ ?

• A. $y=4+x+c{e}^x$
• B. $y=-4-x+c{e}^x$
• C. $y=4-x+c{e}^x$
• D. $y=-4-x-c{e}^{-x}$

Answer 1

The correct option is C $y=4-x+c{e}^x$

Condition given $x+y=\frac{dy}{dx}+5$

$\frac{dy}{dx}=x+y-5$

Let $x+y-5=t$

$\Rightarrow 1+\frac{dy}{dx}=\frac{dt}{dx}\iff \frac{dy}{dx}=\frac{dt}{dx}-1$

$\therefore \frac{dt}{dx}-1=t$

$\Rightarrow \frac{dt}{dx}=t+1$

$\Rightarrow \int \frac{dt}{t+1}=\int dx$

$\Rightarrow \ln (|t+1|)=x+c$

$\Rightarrow t+1=e^{x+c}$

$\Rightarrow x+y-4=k{e}^x$

$y=4-x+k{e}^x$