UPSC 2017 Maths Optional Paper 1 Q5b — Step-by-Step Solution

10 marks · Section B

Orthogonal trajectories (cartesian and polar) · ODEs · asked 7× in 13 yrs · Read the full method →

Question

Suppose that the streamlines of the fluid flow are given by a family of curves $xy=c$ . Find the equipotential lines, that is, the orthogonal trajectories of the family of curves representing the streamlines.

Technique

Form the ODE of the family, swap $y'\to-1/y'$ , integrate the separable result.

Solution

Step 1 — Differential equation of the given family

Differentiate $xy=c$ with respect to $x$ :

y+x\frac{dy}{dx}=0\quad\Longrightarrow\quad \frac{dy}{dx}=-\frac{y}{x}.

The parameter $c$ is already eliminated, so this is the ODE of the streamline family. Its slope at $(x,y)$ is $-y/x$ .

Step 2 — Replace slope by the orthogonal slope

Orthogonal trajectories cross every member at a right angle, so their slope is the negative reciprocal:

\frac{dy}{dx}\;\longrightarrow\;-\frac{1}{(-y/x)}=\frac{x}{y}.

Hence the orthogonal trajectories satisfy

\frac{dy}{dx}=\frac{x}{y}.

Step 3 — Solve (separable)

y\,dy=x\,dx\quad\Longrightarrow\quad \frac{y^2}{2}=\frac{x^2}{2}+\text{const}.

Therefore

\boxed{\,x^2-y^2=k\,}\qquad(k\ \text{arbitrary constant}).

The equipotential lines form the family of rectangular hyperbolas $x^2-y^2=k$ , with the same asymptotes (the axes rotated $45^\circ$ ) as the streamlines $xy=c$ .

Verification

$ python3 -c "import sympy as sp; x,y=sp.symbols('x y'); print(sp.simplify((-y/x)*(x/y)))"
# -1

The product of the two slopes is $(-y/x)(x/y)=-1$ at every point, so the families are orthogonal. ✓ (Both $xy=c$ and $x^2-y^2=k$ are harmonic conjugates — they are the real/imaginary level curves of the analytic function $w=\tfrac12 z^2$ , confirming the flow/potential interpretation.)