Question

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

• A. $y=2(e^x+x–1)$
• B. $y=2(e^x–x–1)$
• C. $y=2(e^x–x+1)$
• D. $y=2(e^x+x+1)$

Answer 1

The correct option is B

$y=2(e^x–x–1)$

Explanation for the correct option:

Finding the equation of curve:

The given slope of the curve is,

$\frac{dy}{dx}=y+2x...\left(i\right)$

Substituting $\mathrm{y}+2\mathrm{x}=\mathrm{z}$

Differentiating it with respect to $x$

$$\begin{gathered} \frac{\mathrm{dy}}{\mathrm{dx}}+2=\frac{\mathrm{dz}}{\mathrm{dx}} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dz}}{\mathrm{dx}}-2...\left(ii\right) \end{gathered}$$

From equations $\left(i\right)\&\left(ii\right)$ we get,

$$\begin{gathered} \Rightarrow \frac{\mathrm{dz}}{dx}-2=\mathrm{z} \\ \Rightarrow \frac{\mathrm{dz}}{\mathrm{dx}}=\mathrm{z}+2 \\ \Rightarrow \frac{\mathrm{dz}}{\mathrm{z}+2}=\mathrm{dx} \\ \Rightarrow \int \frac{\mathrm{dz}}{\mathrm{z}+2}=\int \mathrm{dx}\left[\mathrm{integrating}\right] \\ \Rightarrow \log (\mathrm{z}+2)=\mathrm{x}+\mathrm{c}\left[\because \int \frac{\mathrm{dz}}{\mathrm{x}}=\log \left(\mathrm{x}\right)\right] \\ \Rightarrow \log (\mathrm{y}+2\mathrm{x}+2)=\mathrm{x}+\mathrm{c}[\mathrm{substituting}\mathrm{back}\mathrm{z}=\mathrm{y}+2\mathrm{x}] \\ \Rightarrow \mathrm{y}+2\mathrm{x}+2={\mathrm{e}}^{\mathrm{x}+\mathrm{c}} \\ \Rightarrow \mathrm{y}+2\mathrm{x}+2={\mathrm{e}}^{\mathrm{x}}\cdot {\mathrm{e}}^{\mathrm{c}} \\ \Rightarrow \mathrm{y}=2[{\mathrm{e}}^{\mathrm{x}}-\mathrm{x}-1] \end{gathered}$$

Hence, option (B) is the correct answer