What is the first step in solving a differential equation using the general solution method?

• A. Substituting values into the equation
• B. Finding the roots
• C. Integrating the equation
• D. Graphing the function

Answer: (B)

The first step in solving a differential equation using the general solution method is to find the roots. This involves identifying the characteristic equation associated with a linear differential equation, particularly when dealing with constant-coefficient linear differential equations. The roots obtained from this equation provide crucial insight into the solutions of the differential equation.

Finding the roots allows the identification of different types of solutions (real, distinct; real, repeated; or complex), which then informs the structure of the general solution. This process is fundamental because it lays the groundwork for further steps, such as constructing the complementary solution (homogeneous solution) and, if necessary, the particular solution based on the nature of the roots.

In contrast, substituting values into the equation, integrating the equation, and graphing the function tend to follow after one has established the roots and formed the general solution. These steps are relevant for particular cases or visual interpretation but are not the foundational first step in the context of the general solution method for differential equations.