ODE with modulus sign

Question

How can I solve this ODE ? $$\left|y'(x)\right| +\left|y(x)\right| =0.$$ I can easily solve it without the modulus signs. At present, $y(x)=0$ is the ony solution I can think of.

Answer 1

Suppose that $y\neq 0$. Then $$ \frac{|y'|}{|y|}+1=0\iff \left\lvert\frac{y'}{y}\right\rvert=-1 $$ which is not possible.

Answer 2

Hint. $\left|a\right| +\left|b\right|=0$ if and only if $a=b=0$.