Paper 2, Section II, C

(a) Solve $\frac{d z}{d t}=z^{2}$ subject to $z(0)=z_{0}$. For which $z_{0}$ is the solution finite for all $t \in \mathbb{R}$ ?

Let $a$ be a positive constant. By considering the lines $y=a\left(x-x_{0}\right)$ for constant $x_{0}$, or otherwise, show that any solution of the equation

$\frac{\partial f}{\partial x}+a \frac{\partial f}{\partial y}=0$

is of the form $f(x, y)=F(y-a x)$ for some function $F$.

Solve the equation

$\frac{\partial f}{\partial x}+a \frac{\partial f}{\partial y}=f^{2}$

subject to $f(0, y)=g(y)$ for a given function $g$. For which $g$ is the solution bounded on $\mathbb{R}^{2}$ ?

(b) By means of the change of variables $X=\alpha x+\beta y$ and $T=\gamma x+\delta y$ for appropriate real numbers $\alpha, \beta, \gamma, \delta$, show that the equation

$\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial x \partial y}=0$

can be transformed into the wave equation

$\frac{1}{c^{2}} \frac{\partial^{2} F}{\partial T^{2}}-\frac{\partial^{2} F}{\partial X^{2}}=0$

where $F$ is defined by $f(x, y)=F(\alpha x+\beta y, \gamma x+\delta y)$. Hence write down the general solution of $(*)$.

*Typos? Please submit corrections to this page on GitHub.*