Week 8 Lecture Notes, Lec01 9.5. Linear Equations A first-order linear differential equation is one that can be put into the form: ?? ?? + ?(?)? = ?(?) Where ?(?) and ?(?) are continuous functions on a given interval. We call this the standard form of a linear differential equation and show it with (*). We can also write it as: ?′ + ?(?)? = ?(?) Example. ? 2 ?′ + 2?? = 4 Solution. The RHS seems to be the derivative of ? 2 ? . So: (? 2 ?)′ = 4 → ? 2 ? = 4? + 𝐶 → ? = 4? + 𝐶 ? 2 If the equation was introduced in the standard form: ?′ + 2 ? ? = 4 ? 2 Then we had to first, multiply ? 2 by both sides so that we could write the RHS as the derivative of ? 2 ? .