Paper 2, Section II, C

Differential Equations | Part IA, 2017

(a) Solve dzdt=z2\frac{d z}{d t}=z^{2} subject to z(0)=z0z(0)=z_{0}. For which z0z_{0} is the solution finite for all tRt \in \mathbb{R} ?

Let aa be a positive constant. By considering the lines y=a(xx0)y=a\left(x-x_{0}\right) for constant x0x_{0}, or otherwise, show that any solution of the equation

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

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

Solve the equation

fx+afy=f2\frac{\partial f}{\partial x}+a \frac{\partial f}{\partial y}=f^{2}

subject to f(0,y)=g(y)f(0, y)=g(y) for a given function gg. For which gg is the solution bounded on R2\mathbb{R}^{2} ?

(b) By means of the change of variables X=αx+βyX=\alpha x+\beta y and T=γx+δyT=\gamma x+\delta y for appropriate real numbers α,β,γ,δ\alpha, \beta, \gamma, \delta, show that the equation

2fx2+2fxy=0\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial x \partial y}=0

can be transformed into the wave equation

1c22FT22FX2=0\frac{1}{c^{2}} \frac{\partial^{2} F}{\partial T^{2}}-\frac{\partial^{2} F}{\partial X^{2}}=0

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

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