Solve $ay''+by+c=0$ using separation of variables

I too, like you wondered how to solve these types of problems where an equation with the second derivative and the function are alone given. I usually wondered because these types of problems normally come in physics, for example, equation of simple harmonic motion, $$\ a+{\omega}^{2}x=0$$ Then I realized while solving for these that we can make a substitution and easily solve them. For your problem, it is $$\ a\cdot\frac{{d}^{2}y}{d{x}^{2}}+by+c=0$$ $$\ a\cdot\frac{{d}^{2}y}{d{x}^{2}}=a\frac{d}{dx}\frac{dy}{dx}=a\frac{d(\frac{dy}{dx})}{dy}\cdot\frac{dy}{dx}$$ I think by now it is obvious that the substitution is $$\ u=\frac{dy}{dx}$$ Hence, the problem reduces to $$\ au\frac{du}{dy}+by+c=0$$ This is a differential equation that is easily solvable as the variables can be separated. Can you proceed? Hope it helps!