Question

The equation of one of the curves whose slope at any point is equal to $y + 2x$ is

• A. $y = 2\left(e^{x} + x -1\right)$
• B. $y = 2\left(e^{x} - x -1\right)$
• C. $y = 2\left(e^{x} - x +1\right)$
• D. $y = 2\left(e^{x} + x +1\right)$

Answer 1

Answer: (B)

$y = 2\left(e^{x} - x -1\right)$

Answer 2

Given, $\frac{d y}{d x}=y+2 x\,\,\,...(i)$
Put $y+2 x=z$
$\Rightarrow \frac{d y}{d x}+2=\frac{d z}{d x}$
$\Rightarrow \frac{d y}{d x}=\frac{d z}{d x}-2\,\,\,...(ii)$
From Eqs. (i) and (ii)
$\frac{d z}{d x}-2=z$
$\Rightarrow \int \frac{d z}{z+2}=\int d x$
$\Rightarrow \log (z+2)=x+c$
$\Rightarrow \log (y+2 x+2)=x+c$
$\Rightarrow y+2 x+2=e^{x+c}$
$\Rightarrow y+2 x+2=e^{x} \cdot e^{c}$
$\Rightarrow y=2\left[e^{x}-x-1\right]$ Taking $e^{c}=2$