Q.Solve the following differential equation: $\sec^2 x \tan y \, dx + \sec^2 y \tan x \, dy = 0$

Concept: Homogeneous Differential Equation — the equation is symmetric in $x$ and $y$ and can be separated.

Rewrite as:

Divide both sides by $\tan x \tan y$ (assuming $\tan x, \tan y \neq 0$):

Integrate each side. Recall $\frac{d}{dx}(\tan x) = \sec^2 x$, so:

Thus: …

We are given:

The equation is of the form $M(x,y)\,dx + N(x,y)\,dy = 0$. At first glance, it looks like it might be exact or homogeneous. But notice the symmetry: each term is a product of a function of $x$ and a function of $y$. That’s the hallmark of a separable equation — we can rearrange it so that all $x$ terms are with $dx$ and all $y$ terms with $dy$.

1. Rearranging to separate variables

We want to isolate $dx$ and $dy$ on opposite sides. Move the second term to the right:

Now divide both sides by $\tan x \tan y$ (assuming $\tan x \neq 0$, $\tan y \neq 0$ — we’ll handle singular cases later):

Each side is now a function of a single variable. The equation is separable.

2. Integrating both sides

We integrate:

Notice that $\frac{d}{dx}(\tan x) = \sec^2 x$. So the numerator is exactly the derivative of the denominator. This suggests a simple substitution: let $u = \tan x$, then $du = \sec^2 x \, dx$. Similarly for $y$.

Thus:

where $u = \tan x$, $v = \tan y$.

Integrating:

So:

Exponentiate both sides:

Let $e^{C_1} = C$ (a positive constant). Then:

Since $\pm C$ is just another constant (call it $C$), we write:

3. Checking the lost solutions …

Method: Separating a symmetric $dx$–$dy$ equation

Use this when the equation is written symmetrically as $M\,dx + N\,dy = 0$ and each coefficient factors into an $x$-part times a $y$-part, so a single division separates the variables.

Steps

Step 1: Divide by the mixing factor

Divide the whole equation by the product that appears in both terms (here, $\tan x\,\tan y$) so that each term ends up with only one variable.

Step 2: Recognise the resulting quotients

Each side becomes a form like $\dfrac{\sec^2 x}{\tan x}\,dx$, whose integral is $\log|\tan x|$ — a "$f'/f$" pattern giving a logarithm. …

Common Mistakes

Mistake 1: Not dividing by the full mixing factor $\tan x\tan y$

Why it's wrong: unless you divide by both $\tan x$ and $\tan y$, the variables stay entangled and the equation will not separate. Correct approach: divide the whole equation by $\tan x\,\tan y$ so each term holds one variable.

Mistake 2: Missing the $\dfrac{\sec^2}{\tan}$ logarithm pattern …