Linear Differentiation Question

Question

Find the general solution to the following differential equation.

$$\frac{dy}{dx} = \frac{8y}{9} $$

These are the possible answers given...

(A) $y = e^{(9/8)x} + C$

(B)  $y = x^{8/9} + C$

(C)  $y = Ce^{(8/9)x}$

(D)  $y = Cx^{8/9}$

(E)  $y = Ce^{(9/8)x}$

(F)  $y = Cx^{9/8}$

(G)  $y = e^{(8/9)x} + C$

(H)  $y = x^{9/8} + C$

Answer 1

This is a separable ODE. Try separating the variables by dividing both sides by $y$, then integrating both sides with respect to $x$:

$$\frac{dy}{dx}=\frac{8y}{9}$$ $$\frac{1}{y}\cdot \frac{dy}{dx}=\frac{8}{9}$$ $$\int \frac{1}{y}~dy=\int \frac{8}{9}~dx$$ Can you continue?

Answer 2

We have,

$\frac 1y dy = \frac 89 dx$

Now on integration,

$\log y = \frac 89 x + \log c$

$\log y - log c = \frac 89 x$

Hope you can proceed now.