We need some mathematical tricks to solve such a differential equation where \(P\left(t\right)\ne0\). This is an inhomogeneous equation, because if \(x_1\left(t\ \right)\) is a solution and if \(x_2\left(t\right)\) is another solution, then \(x_3=\left(x_1+x_2\right)\) is not a solution.
What we can do is to consider an associated homogeneous equation of the form: