Solve the differential equation dy/dx + ay = b, where a and b are constants.

Step-by-step solution

1. 1

   Identify the standard form. The equation is already $y'+P(x)y=Q(x)$ with constant coefficients:

   $P(x)=a,\qquad Q(x)=b$

2. 2

   Compute the integrating factor.

   $\mu(x)=e^{\int a\,dx}=e^{ax}$

3. 3

   Multiply through and collapse the left side.

   $e^{ax}\frac{dy}{dx}+ae^{ax}y=be^{ax}\quad\Longrightarrow\quad\frac{d}{dx}\left(e^{ax}y\right)=be^{ax}$

4. 4

   Integrate, assuming $a\neq 0$.

   $e^{ax}y=\frac{b}{a}e^{ax}+C\quad\Longrightarrow\quad y=\frac{b}{a}+Ce^{-ax}$

   The division by $a$ is exactly why $a=0$ needs its own treatment.

5. 5

   Handle $a=0$. The equation degenerates to $\tfrac{dy}{dx}=b$, whose solution is a straight line:

   $y=bx+C$

6. 6

   Verify and interpret. For $a\neq 0$: $y'=-aCe^{-ax}$ and $ay=b+aCe^{-ax}$, so $y'+ay=b$ $\checkmark$ (numerically confirmed at $a=1.3$, $b=2.1$, $C=0.7$, $x=0.6$). The constant $\tfrac{b}{a}$ is the equilibrium: if $a>0$ every solution decays towards it, which is the shape behind Newton's law of cooling and RC circuit charging.