6.3 Initial conditions

In the equation $y^{\prime}=f(x)$, $y$ is the dependent variable and $x$ is the independent variable. In many physical applications, the independent variable is denoted $t$ and represents time. Then the equation

describes the rate of change of $z$, which here depends only on time. Often we want to find a solution $z(t)$ to the equation which has a certain value $\xi\in{\mathbb{R}}$ at a particular time $t_{0}$, i.e. such that $z(t_{0})=\xi$. This is known as an initial condition. In the previous slide, an initial condition $y(x_{0})=\xi$ uniquely specifies the value of the constant $c$ (unless $x_{0}=2$).

Solution. We have $\log|1-2|+c=1$, thus $c={1.}$