← 2019 Paper 2

UPSC 2019 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 of the family of surfaces given by the following expression:

Technique

Eliminate the arbitrary function $\psi(u,v)=0$ by setting the Jacobian $\partial(u,v)/\partial(x,y)=0$ (with $z=z(x,y)$). Standard “function of two combinations” method.

Solution

Setup. Write $u=x^2+y^2+2z^2$ and $v=y^2-2zx$, so the relation is $\psi(u,v)=0$ with $z=z(x,y)$, $p=z_x$, $q=z_y$. Differentiating $\psi(u,v)=0$ partially eliminates the arbitrary function $\psi$ via the Jacobian condition.

Step 1 — Differentiate $\psi(u,v)=0$ w.r.t. $x$ and $y$

For a non-trivial $(\psi_u,\psi_v)$ the determinant must vanish:

Step 2 — Compute the total derivatives (treating $z=z(x,y)$)

Step 3 — Impose the Jacobian condition

Divide by $2$ and expand:

The cross terms $-4xzpq$ and $+4xzpq$ cancel:

Divide by $2$ and group $p,q$:

So

Answer