JEE 2023 — Mathematics PYQ

Given that $\frac{dy}{dx}=y+7$

$$\Rightarrow \int \frac{dy}{y+7}=\int dx\Rightarrow {\log }_e(y+7)=x+c$$

$y_1(0)=0\Rightarrow {\log }_{e}7=0+c$

$$\Rightarrow c={\log }_{e}7\Rightarrow {\log }_e(y+7)=x+{\log }_{e}7$$

$$\Rightarrow {\log }_e\left(\frac{y+7}{7}\right)=x$$

$$\text{or }y+7=7e^x\dots (1)$$

Also, $y_2(0)=1\Rightarrow {\log }_{e}8=1+c$

$$\Rightarrow c={\log }_e\left(\frac{8}{e}\right)\Rightarrow {\log }_e(y+7)=x+{\log }_e\left(\frac{8}{e}\right)$$

$$\Rightarrow {\log }_e\left(\frac{(y+7)e}{8}\right)=x$$

$$\Rightarrow (y+7)e=8e^x\dots (2)$$

From (1) and (2)

$$7e^x\times e=8e^x\Rightarrow e=\frac{8}{7}\text{ which is not true.}$$

Hence, there is no point.