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?
Best Answer
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): $$xdy+ydx = 0 \Leftrightarrow d(xy) = 0 \implies xy = C$$