Q.The solution of the differential equation 𝑥𝑑𝑥 + 𝑦𝑑𝑦 = 0 represents a family of
(A) straight lines
(B) parabolas
(C) Circles
(D) Ellipses

The key idea is that the given differential equation can be integrated directly to obtain the equation of a curve.

Step 1: Rewrite the equation:

$x \, dx + y \, dy = 0$

Step 2: Integrate both sides:

$\int x \, dx + \int y \, dy = \int 0$

$\frac{x^2}{2} + \frac{y^2}{2} = C$

Step 3: Multiply through by 2:

$x^2 + y^2 = 2C$ …

Why this approach works

When you see a differential equation written in the form $x\,dx + y\,dy = 0$, the first thing to notice is that the variables are already separated — each term involves only one variable paired with its own differential. That means we can integrate term by term directly, without any rearrangement.

The key insight: $x\,dx$ integrates to $\frac{x^2}{2}$, and $y\,dy$ integrates to $\frac{y^2}{2}$. Summing them gives $\frac{x^2 + y^2}{2} = \text{constant}$, which is exactly the equation of a circle centered at the origin. The constant determines the radius.

Step-by-step solution

1. Separate and integrate The equation is already separated:

Integrate both sides:

This gives:

where $C_1$ is an arbitrary constant of integration.

2. Simplify the constant Multiply through by 2:

Let $c = 2C_1$, which is still an arbitrary constant (any real number, usually taken as positive for a real circle). So: …

Method: Identifying the family of curves from a directly-integrable first-order DE

Use this whenever a first-order differential equation is already written as a sum of one-variable-times-its-own-differential terms — you integrate directly and read off the geometry.

Steps

Step 1: Check whether the variables are already separated

An equation of the form $f(x)\,dx + g(y)\,dy = 0$ has each variable paired only with its own differential. No rearrangement is needed — you may integrate term by term.

Step 2: Integrate each term

Always add a single arbitrary constant $C$. …

Common Mistakes

Mistake 1: Reading "$x\,dx+y\,dy=0$" as a straight line because it looks linear

Why it's wrong: the presence of $dx$ and $dy$ means this is a differential equation to be integrated, not a linear relation $ax+by=0$. Correct approach: integrate to get $x^2+y^2=c$, which is a circle.

Mistake 2: Dropping the constant or mishandling the factor of 2 …