An example Imagine we’re looking into a system that has the following sort of form $$ \ddot{y}(t) + c \dot{y}(t) + k y(t) = x_{1}(t) + x_{2}(t) $$ where \( y \) is our output and there are two inputs, \( x_{1} \) and \( x_{2} \). For the purpose of this example, we don’t know exactly what \( c \) and \( k \) are - the specific system we have is a bit of a mystery....
The Initial Value Problem In engineering, we often encounter systems that evolve over time, such as circuits, mechanical systems, or chemical reactions. These systems are best described using differential equations. For example, Newton’s law of cooling states: $$ \dfrac{dT}{dt} = -k(T - T_{surr}) $$ where T is the temperature of some point, k is a proportionality constant, and Tsurr is the temperature surrounding the point of interest. To solve this equation is to find a solution for the temperature, T, over time....