1. The problem is to understand what stochastic differential equations (SDEs) are and how they are used. 2. A stochastic differential equation is a differential equation in which one or more terms are stochastic processes, leading to a solution that is itself a stochastic process. 3. The general form of an SDE is: $$ dX_t = \mu(X_t,t) dt + \sigma(X_t,t) dW_t $$ where $X_t$ is the stochastic process, $\mu$ is the drift coefficient, $\sigma$ is the diffusion coefficient, and $W_t$ is a Wiener process or Brownian motion. 4. The term $\mu(X_t,t) dt$ represents the deterministic part of the change in $X_t$, while $\sigma(X_t,t) dW_t$ represents the random fluctuations. 5. To solve SDEs, methods like Itô calculus are used, which extend traditional calculus to stochastic processes. 6. Understanding SDEs requires knowledge of probability theory, stochastic processes, and differential equations. 7. Applications include physics, finance (modeling stock prices), biology, and engineering. Final answer: Stochastic differential equations model systems influenced by random noise and are expressed as $$ dX_t = \mu(X_t,t) dt + \sigma(X_t,t) dW_t $$ where $W_t$ is Brownian motion.