How to find solution of the following differential equation

Question

How to find solution of the following differential equation?

$\frac{\partial\phi}{\partial t} = a \frac{\partial\phi}{\partial x}$, where $a$ is constant.

Answer 1

$$\frac{\partial\phi}{\partial t} = a \frac{\partial\phi}{\partial x}$$ General solution : $$\phi(x,t)=F(x+at)$$ $F(X)$ is an arbitrary function.

Examples :

With $F(X)=X\quad$ a solution is $\quad \phi(x,t)=x+at$ .

With $F(X)=X+C\quad$ a solution is $\quad \phi(x,t)=x+at+C$ .

With $F(X)=X^5\quad$ a solution is $\quad \phi(x,t)=(x+at)^5$ .

With $F(X)=\sin(X+c)\quad$ a solution is $\quad \phi(x,t)=\sin(x+at+c)$ .

With $F(X)=5e^{X}+\frac{3}{X}\quad$ a solution is $\quad \phi(x,t)=5e^{x+at}+\frac{3}{x+at}$ .

etc.

You can see that they are an infinity of solutions if no condition is specified.

If some boundary or initial conditions are specified in the wording of the problem, it is often possible to determine what is the function $F$. Then putting it into the general solution leads to one solution which satisfies both the PDE and the boundary condition.