Solve for $t$: $ e^{-2t} + 2t = 4 $

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How do we do this problem for other values of the constant, say 300 or -1000? Is there a general way to solve such questions? (Looking for a way to solve this with pen and paper.)

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Numerically. The exact solution involves the Product Log (Lambert-W) function.

By hand, you can use Newton's method or similar.

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$$e^{-2t}+2t=4$$ $$e^{-2t}=4-2t$$ $$e^{2t}=\frac{1}{4-2t}$$ $$(4-2t)e^{2t}=1$$ $$(2t-4)e^{2t-4}=-\frac{1}{e^4}$$ $$2t-4=W\left(-\frac{1}{e^4}\right)$$ $$t=\frac12W\left(-\frac{1}{e^4}\right)+2$$

This solution uses the Lambert W-function.