1487.7=6.5x-12.3(e^(-1.5x)) how to solve for x? Differential equation?

You could use a graphical method or a numerical method.

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1487.7  = 6.5x - 12.3*e^-1.5x

 

Must have x>0 as the RHS is negative for x<0.

 

As x gets larger than zero the exponential term gets smaller and smaller, so an initial estimate for x might be to assume the exponential term is so small it's negligible. This will result in an initial guess for x as

 

x0 = 1487.7/6.5

$${\mathtt{x0}} = {\frac{{\mathtt{1\,487.7}}}{{\mathtt{6.5}}}} \Rightarrow {\mathtt{x0}} = {\mathtt{228.876\: \!923\: \!076\: \!923\: \!076\: \!9}}$$ 

 

 

With this value for x, how big is the exponential term?

$${\mathtt{12.3}}{\mathtt{\,\times\,}}{{\mathtt{e}}}^{{\mathtt{\,-\,}}\left({\mathtt{1.5}}{\mathtt{\,\times\,}}{\mathtt{228.866\: \!823\: \!076\: \!923\: \!076\: \!9}}\right)} = {\mathtt{0}}$$

 

It's not really 0, it's actually about 2*10^-14, but it is small enough to ignore for most purposes!

 

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Thanks! Good answer! 

 

I wonder if there is a general way of solving this kind of equation without understanding that the exponential term is going to be 0. 

 

Like an equation where the expontential term is big enough to be considered.