# Tricky Separable Differential Equation Please guide me: y′+ay+b=0 (a not zero) is supposed to be separable and has solution y = ce^(-ax) - b/a Here is my start to this problem: (dy)/(dx) + ay = -b is as far as I can go with this. How should I go about separating x and

Tricky Separable Differential Equation
${y}^{\prime }+ay+b=0$
(a not zero) is supposed to be separable and has solution
$y=c{e}^{-ax}-\frac{b}{a}$
Here is my start to this problem:
$\frac{dy}{dx}+ay=-b$ is as far as I can go with this. How should I go about separating x and
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autarhie6i
$\frac{dy}{dx}+ay+b=0$
$\frac{dy}{dx}=-ay-b$
$\frac{dy}{ay+b}=-dx$
Now, can you solve this?
###### Not exactly what you’re looking for?
Bruno Thompson
We can take the differential equation
${y}^{\prime }+ay+b=0$
and write
${y}^{\prime }+ay=-b$
then multiply by ${e}^{ax}$
${e}^{ax}{y}^{\prime }+a{e}^{ax}y=-5{e}^{ax}$
Now the term on the left becomes
$\left(y{e}^{ax}{\right)}^{\prime }=-5{e}^{ax}.$
Next antidifferentiate and solve for y.