Why the exponential function is the ultimate mathematical tool for solving equations
In short: In the world of differential equations, some functions are simply easier to work with than others. When you face a system experiencing both internal growth and external input, the exponential function provides a unique, elegant shortcut to finding the complete solution.
In his MIT course on differential equations, Professor Gilbert Strang explores the behavior of first-order linear equations subject to external forces. Specifically, he examines a scenario where a system experiences internal exponential growth—such as interest being added to a balance—alongside an external source term, like continuous deposits. This setup creates a dynamic where both the internal mechanism and the outside input are driving the system's change.
The exponential function, $e^{st}$, is uniquely suited for this problem because its derivative is itself. This property allows for a method called finding the 'particular solution.' By substituting an assumed exponential form into the equation, one can solve for a specific coefficient, $Y$. For a system where the rate of change $dy/dt$ equals $ay$ plus an input of $e^{st}$, the particular solution reveals itself to be $Y$ multiplied by $e^{st}$, where $Y$ is exactly $1 / (s - a)$.
However, a single particular solution is rarely enough to describe a real-world system, as it may not satisfy the 'initial condition'—the state of the system at time $t=0$. To find the complete solution, one must add the 'null' or homogeneous solution, which represents the system's behavior when the external source is zero. The final, general solution is a combination of these two exponentials, adjusted by a constant $C$ to ensure the math matches the starting value, $y(0)$.
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Sources
- Summary by Intellect · source: Response to Exponential Input
- Also: Response to Exponential Input | Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler | Mathematics | MIT OpenCourseWare
- Also: Response to Exponential Input | MIT Learn
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