← 2020 Paper 2

UPSC 2020 Maths Optional Paper 2 Q5a — Step-by-Step Solution

10 marks · Section B

Family of surfaces · PDEs · asked 7× in 13 yrs · Read the full method →

Question

Form a partial differential equation by eliminating the arbitrary functions $f(x)$ and $g(y)$ from $z=y\,f(x)+x\,g(y)$ and specify its nature (elliptic, hyperbolic or parabolic) in the region $x>0,\ y>0$.

Technique

Eliminate two arbitrary one-variable functions by forming $xp+yq$ and identifying the mixed derivative $s$; classify via discriminant $B^2-AC$ of the principal part.

Solution

We have two arbitrary functions, so we expect a second-order PDE.

Step 1 — First-order partial derivatives

Step 2 — A mixed second derivative

Step 3 — Eliminate $f,g$

Form the combination $x\,p+y\,q$:

The last bracket is exactly $z$, and $f'(x)+g'(y)=s$. Hence

Rearranging,

This is the required PDE (all arbitrary functions eliminated).

Step 4 — Classification

Write the principal (second-order) part as

Here $z_{xx}$ and $z_{yy}$ are absent, so $A=0,\ C=0$, and the coefficient of $z_{xy}$ is $xy$, giving $2B=xy$, i.e. $B=\tfrac{xy}{2}$.

The discriminant is

In the region $x>0,\ y>0$ we have $x^2y^2/4>0$, so $B^2-AC>0$ everywhere there.

Answer