Let y = y _ 1 (x) and y = y _ 2 (x) be the solution curves the...

We have been given that$\dfrac{dy}{dx} = y + 7 ⇒ \dfrac{dy}{y + 7} = dx$

$⇒ ln | y + 7 | = x + c$

$⇒ | y + 7 | = k . {e}^{x}$

$⇒ y = k \cdot {e}^{x} - 7$

$⇒ {y}_{1}\left(0\right) = 0 ⇒ k = 7 ⇒ {y}_{1}\left(x\right) = 7\left({e}^{x} - 1\right)$

$⇒ {y}_{2}\left(0\right) = 1 ⇒ 1 = k - 7 ⇒ 8 = k$

$⇒ {y}_{2}\left(x\right) = 8{e}^{x} - 7$

$⇒ {y}_{1}\left(x\right) = {y}_{2}\left(x\right) ⇒ 8{e}^{x} - 7 = 7{e}^{x} - 7$

Hence, No point of intersection.

This is the required option.