We have been given that$\frac{dy}{dx}=y+7\Rightarrow \frac{dy}{y+7}=dx$

$\Rightarrow \ln |y+7|=x+c$

$\Rightarrow |y+7|=k.e^x$

$\Rightarrow y=k·e^x-7$

$\Rightarrow y_1\left(0\right)=0\Rightarrow k=7\Rightarrow y_1\left(x\right)=7\left(e^x-1\right)$

$\Rightarrow y_2\left(0\right)=1\Rightarrow 1=k-7\Rightarrow 8=k$

$\Rightarrow y_2\left(x\right)=8e^x-7$

$\Rightarrow y_1\left(x\right)=y_2\left(x\right)\Rightarrow 8e^x-7=7e^x-7$

Hence, No point of intersection.

This is the required option.