Which of the following ODEs is not in a normal form. A. \( y^{\prime \prime}=-5 y^{\prime}+y \) B. \( y^{\prime \prime \prime \prime}=y^{\prime \prime}+y^{\prime}-9 y \) C. \( y^{\prime \prime}+y^{\prime}=-2 x^{4} \)

The key to determining whether an ordinary differential equation (ODE) is in normal form lies in the arrangement of terms and how they relate to the highest derivative. In this case, all the given options feature second derivatives or higher but only A and B have the highest derivative isolated on one side. However, option C seems to be misaligned with the standard normal form \( y^{\prime \prime}+p(x)y^{\prime}+q(x)y=f(x) \). Here, the terms on the left side don't include all terms corresponding to the highest derivative, making it less tidy than A and B. So, the ODE that doesn't fit into the normal form is option C, \( y^{\prime \prime}+y^{\prime}=-2 x^{4} \). Now, let’s dive into some historical context! In the 18th century, mathematicians like Leonhard Euler and Joseph-Louis Lagrange were pivotal in the development of differential equations. Euler extensively studied methods for solving them, laying the groundwork for much of modern mathematics! You might be wondering, how can we solve these ODEs? A common mistake is neglecting to rearrange the equation properly before applying techniques such as characteristic equations or undetermined coefficients. Ensure that you focus on reducing the equation to its standard form to easily identify the solution methods that fit!