Solving differential equation $xdy + ydx = 0$ Solve following equation

$$xdy + ydx = 0$$

My process: group variables 

$$xdy = -ydx \ \ \ ; \ \ \ \frac{x}{dx} = -\frac{y}{dy}$$

Integrate

$$\int{\frac{dx}{x}} = \int{\frac{-dy}{y}}$$
$$\ln |x| + K = - \ln |y| + K$$

Solution 
$$\ln|xy| = K$$

However my textbook gives answer:

$$xy = K$$

Where am I going wrong?
Your approach is alright. Note that if $\ln A = k$, then $A = e^k$ and if $k$ is a constant, then so is $e^k$ and you could rename it to (for example) $K$.

Other approach (using the product rule):
$$xdy+ydx = 0 \Leftrightarrow d(xy) = 0 \implies xy = C$$
$$
Xdy + ydx =0
xdy = -- ydx
1/y dy = -- 1/x dx
$$

Integrating both sides;
$$
log |y| = -- log |x| + log |C| $$$$
log |y|  + log |x| = log |C| $$$$
log |xy| = log |C|
$$
Comparing respective parts;
$$xy = C$$$$
y = C/x 
$$, 
which is required general solution..