Trying to differentte $\ln(|2+f(x)|)=2+e^{x*x}$

Question

I am trying to solve this differential

$\ln(|2+f(x)|)=2+e^{x*x}$

so far I did this much; $$ \ln(|2+f(x)|)=2+e^{x*x}\\ |2+f(x)|=e^{2+e^{x*x}}\\ \text{now I have two situations/solutions, because of absolute value}\\ |2+f(x)|>=0\\ \text{where I got that}\\ f(x)=e^{2+{e^{x*x}}}-2\\ |2+f(x)|<0\\ \text{where I got that}\\ f(x)=-e^{2+{e^{x*x}}}-2\\ \text{and this is strange, because the solution is $f(x)=-e^{2+{e^{x*x}}}+2$} $$

I am doing something wrong or are the solutions in the book wrong??? Thanks

Answer 1

You're correct,though you could check if they wrote $\ln(|2+f(x)|)=2+e^{x*x}$ or $|\ln(2+f(x))|=2+e^{x*x}$ $$\ln(|2+e^{2+e^{x*x}}-2|)\\\ln(|e^{2+e^{x*x}}|)\\\ln(e^{2+e^{x*x}})=2+e^{x*x}\\\ln(|2-{e^{2+e^{x*x}}-2|)}\\\ln(|-e^{2+e^{x*x}}|)\\\ln(e^{2+e^{x*x}})=2+e^{x*x}$$