Multiply by $\displaystyle x^6$
$\displaystyle \frac{dy}{dx}=x^6(19+9y)$
Divide by $\displaystyle 19+9y$
$\displaystyle \frac{dy}{dx(19+9y)}=x^6$
Integrate
$\displaystyle \int\frac{dy}{19+9y}=\int x^6 dx$
$\displaystyle \int \frac{dy}{19+9y}=\int\frac{dy}{9(\frac{19}{9}+y)}= \frac{1}{9}\ln(\frac{19}{9}+y)+C$
$\displaystyle \int x^6dx=\frac{x^7}{7}+\bar{C}$
$\displaystyle \frac{1}{9}\ln(\frac{19}{9}+y)+C=\frac{x^7}{7} + \bar{C} \rightarrow \ln(\frac{19}{9}+y)=\frac{9x^7}{7}+D$
Taking exponentials
$\displaystyle \frac{19}{9}+y=e^{\frac{9x^7}{7}+D}=Ke^{\frac{9x^7 }{7}}$
So
$\displaystyle \boxed{y=Ke^{\frac{9x^7}{7}}-\frac{19}{9}, K \in \mathbb{R}}$
Checking the answer:
$\displaystyle \frac{dy}{dx}=Ke^{\frac{9x^7}{7}}9x^6$
Noting that
$\displaystyle Ke^{\frac{9x^7}{7}}=y+\frac{19}{9}$
$\displaystyle \frac{dy}{dx}=\left(y+\frac{19}{9}\right)9x^6=(9y+ 19)x^6$
As requested