SDE: What is a Stochastic Differential Equation?

 Consider the population growth model
$$\frac{dN}{dt}=a(t)N(t),\ N(0)=N_0 \tag{1}$$
where $N(t)$ is the size of a population at time $t$ and $a(t)$ is the relative growth rate at time $t$. If $a(t)$ is completely known, one can easily solve (1). In fact, the solution would be $N(t)=N_0\exp\left(\int_0^t a(s)ds\right)$. Now suppose that $a(t)$ is not completely known but it can be written as $a(t)=r(t)+\mbox{noise}$. We do not know the exact behavior of noise but only its probability distribution. Such a case equations like (1) is called a stochastic differential equation. More generally, a stochastic differential equation can be written as
$$\frac{dX}{dt}=b(X(t))+B(X(t))\xi(t)\ (t>0),\ X(0)=x_0,\ \tag{2}$$
where $b: \mathbb{R}^n\longrightarrow\mathbb{R}^n$ is a smooth vector field and $X: [0,\infty)\longrightarrow\mathbb{R}^n$, $B: \mathbb{R}^n\longrightarrow\mathbb{M}^{n\times m}$ and $\xi(t)$ is an $m$-dimensional white noise. If $m=n$, $x_0=0$, $b=0$ and $B=I$, then (2) turns into
$$\frac{dX}{dt}=\xi(t),\ X(0)=0 \tag{3}$$
The solution of (3) is denoted by $W(t)$ and is called the $n$-dimensional Wiener process or Brownian motion. In other words, white noise $\xi(t)$ is the time derivative of the Wiener process. Replace $\xi(t)$ in (2) by $\frac{W(t)}{dt}$ and divide the resulting equation by $dt$. Then we obtain
$$dX(t)=b(X(t))dt+B(X(t))dW(t),\ X(0)=x_0 \tag{4}$$The stochastic differential equation (4) is solved symbolically as
$$X(t)=x_0+\int_0^tb(X(s))ds+\int_0^tb(X(s))dW(s) \tag{5}$$for all $t>0$. In order to make sense of $X(t)$ in (5) we will have to know what $W(t)$ is and what the integral $\int_0^tb(X(s))dW(s)$, which is called a stochastic integral, means.