Find the equation for the streamline following the vector field $$v(x,y) = 8x\textbf{i} + (9x+16y)\bf{j}$$ which passes through the point $(1,18)$. Write the equation in the form $y = f(x)$.
I started out with $$\frac{dx}{8x} = \frac{dy}{9x+16y} \implies \frac98 + \frac{2y}x = \frac{dy}{dx} $$ Not sure how to continue from this.
you got linear ODE
of form $\dfrac{dy}{dx}+Py=Q$
$P=\dfrac{-2}{x}\implies I.F(integrating factor)=\dfrac{1}{x^2}$
$\dfrac{y}{x^{2}}=\dfrac{9}{8}.\displaystyle\int \dfrac{1}{x^2} dx+C$
$\dfrac{y}{x^2}=-\dfrac{9}{8x}+C$
since, it pass through$(1,18)$
$\dfrac{y}{x^2}=-\dfrac{9}{8x}+\dfrac{153}{8}$