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Converging and Diverging Differentials


Thomas Hayes:
Is this true?

Assume you have a first order differential of the form $\frac{dy}{dx} = ay + b$
If $a > 0$, then the solutions will diverge
If $a < 0$, then the solutions will converge

Victor Ivrii:
I edited your post, making mathematical formulae properly displayed. Avoid typing in "all caps".  Instead you can use bold, italic, underline and typewriter styles.

The answer is positive, if "converging" and "diverging" means "has a limit as $t\to +\infty$ and "has a limit as $t\to -\infty$" respectively. The answer is trivial, as long as you can solve this equation explicitly, which will be taught Week 2.

Zhanhao Ye:
I think that's true. By isolating the variables X and Y, we can get an differential equation of 1/(ay+b) dy = dx. Then, take the integral on both side, a log function of y can be obtained. After simplifying the equation, we can have an exponential function with the coefficient of 'a'. Convergence or divergence depends on the sign of 'a'.


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