\begin{align} y &= \frac{a}{b} \left(1 - e^{-bx} \right) \\ \implies y’ &= \frac{a}{b} \cdot b e^{- bx} \\ &= ae^{- bx} \\ \implies y’ &= a \left(1 - \frac{by}{a} \right) \\ &= a - by \end{align}

Perfectly logical. No problem at all.

But I have no clue at all how to go in reverse from

$$y’ = a - by$$

to the solution $y$. I think I need to find an integrating factor, but how? I also thought about a characteristic equation. Is that the way to go?

This is probably an obvious question, but all I know about differential equations, I picked up on my own rather than from a course or a text. I understand separable equations, exact equations, and characteristic equations, but finding integrating factors is terra incognita.

  • 2
    $\begingroup$ $y'=a-by$ is separable $$\frac{dy}{a-by}= dx \implies \ln (a - by) = - b x + C \implies \dots$$ or can be integrated using an integrating factor as you stated $$y' + by = a \implies \text{IF} = \exp \left(\int b dx \right) = \exp \left( bx\right) \implies e^{bx} y' + be^{bx}y = ae^{bx} \implies \dots$$ or can be solved by finding the homogeneous (by making the ansatz $y_{h} = e^{\lambda x}$ and deriving the characteristic equation) and particular ($y_{b}=A=\text{constant}$) solutions. $\endgroup$
    – mattos
    Jun 26 at 20:31
  • $\begingroup$ Thank you very much. I had a blind spot on the separabilty; I would have wasted hours on that. I shall have to think about the integrating factor logic. But again, thank you. $\endgroup$ Jun 26 at 20:43
  • $\begingroup$ Oh, and thanks as well to the edits to put the question in LaTeX. I tried quite a few delimiters to invoke it, but none worked. $\endgroup$ Jun 26 at 21:08
  • 1
    $\begingroup$ The integrating factor is simply $e^{bx}$ $\endgroup$
    – MtGlasser
    Jun 26 at 21:30
  • $\begingroup$ Ahh, the coefficient of y becomes the coefficient of x in the exponential. Always obvious once the light bulb turns on. Thank you $\endgroup$ Jun 26 at 22:57

If you have an ODE of the form $y' + fy = g,$ where $f$ and $g$ are given functions, then the integrating factor is $e^{F},$ where $F$ is a primitive function of $f,$ i.e. $F'=f$: $$ e^F(y'+fy) = (e^Fy)'^. $$


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