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.)
2025-01-13 07:56:36.1736754996
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Solve for $t$: $ e^{-2t} + 2t = 4 $
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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.
Numerically. The exact solution involves the Product Log (Lambert-W) function.
By hand, you can use Newton's method or similar.