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January 3rd, 2019, 08:02 PM  #1 
Newbie Joined: Jan 2019 From: Europe Posts: 2 Thanks: 0  Solve an equation with two variables
Hello, I am trying to solve this equation, but I can't find any way to do that. The solution has to be a=b=0, but how can I prove that? Thank you in advance. 2e^(a^2)+ln(b^2+1)=2 
January 3rd, 2019, 09:00 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,105 Thanks: 1907 
If a and b are real, 2e^(a^2) is at least 2 and ln(b^2 + 1) is at least 0. Hence 2e^(a^2) + ln(b^2+1) = 2 implies 2e^(a^2) = 2 and ln(b^2+1) = 0. Solving those equations gives a = 0 and b = 0. If a and b need not be real, there are other solutions. 
January 3rd, 2019, 09:00 PM  #3  
Senior Member Joined: May 2016 From: USA Posts: 1,251 Thanks: 519  Quote:
$a = 0 = b \implies \{2e \text {^}(a^2)\} + ln(b^2 + 1) = \{2e \text {^} 0^2\} + ln(0 + 1) =$ $(2 * e^0)+ ln(1) = (2 * 1) + 0 = 2.$ That is a proof. EDIT: skipjack has provided a proof that is the only real solution. Last edited by JeffM1; January 3rd, 2019 at 09:04 PM.  

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