3.1.2 Existence and uniqueness for linear first order ode in \(y\)

These are ode’s in the form

\[ y^{\prime }+p\left ( x\right ) y=q\left ( x\right ) \]

The theorem says that if both \(p\left ( x\right ) ,q\left ( x\right ) \) are continuous at \(x_{0}\) then solution exists and is unique. Notice that now we do not check on \(y_{0}\) but only on \(x_{0}\). We get both existence and uniqueness all in one test. If either \(p\) or \(q\) are not continuous, then no guarantee solution exist or be unique.