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Can we estimate the following relationship using linear regression. Here, $\beta_1, \beta_2 $ and $\beta_3$ are parameters.

$$ y = \beta_1\exp(x) + \beta_2\exp(-x) + \beta_3$$

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This is just multilinear regression.

You have $n$ data points $(x_i,y_i)$. For each point, define $u_i=e^{x_i}$ and $v_i=e^{-x_i}$

$$y = \beta_1\,u + \beta_2\,v + \beta_3$$

Linearity is related to the parameters, not to the functions.

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