Question

Determine a condition on $A$ and $B$ for which the equation
$A\cosh x+B\sinh x=0$ has at least one real solution.

Answer 1

$A\cosh x+B\sinh x=0$

$\Rightarrow A\left(\frac{e^x+e^{-x}}{2}\right)+B\left(\frac{e^x-e^{-x}}{2}\right)=0$

$\Rightarrow e^x(A+B)+e^{-x}(A-B)=0$

$\Rightarrow e^{2x}=\frac{B-A}{B+A}$

As $e^{2x}\ge 0\Rightarrow$for the equation to have at least one real solution $\frac{B-A}{B+A}=0$

$\Rightarrow A=B$ is the condition