Q.The differential equation $y\frac{dy}{dx}+x=c$ represents:
(A) Family of hyperbolas
(B) Family of parabolas
(C) Family of ellipses
(D) Family of circles

The key idea is to rewrite the differential equation in a form that reveals the geometric shape of its solution curves.

Step 1: Rewrite the equation.

$y\frac{dy}{dx} + x = c \implies y\,dy + x\,dx = c\,dx$

Step 2: Integrate both sides.

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

$\frac{y^2}{2} + \frac{x^2}{2} = cx + k$, where $k$ is the constant of integration.

Step 3: Rearrange into standard form.

Multiply through by 2: $x^2 + y^2 = 2cx + 2k$ …

We are asked: what family of curves does $y\frac{dy}{dx}+x=c$ represent? The key is to recognise that this is a first-order differential equation that can be solved by separating variables — but more importantly, the structure hints at a relation between $x$ and $y$ that is symmetric and quadratic.

Let’s rewrite it cleanly:

Bring $x$ to the other side:

Now multiply both sides by $dx$ (treating $dy/dx$ as a ratio, which is valid here):

This is a separable differential equation. We can integrate both sides directly.

Step 1: Integrate both sides

where $C$ is the constant of integration.

Step 2: Rearrange into a recognisable form

Multiply through by 2:

Bring all terms to one side:

Step 3: Complete the square in $x$

We have $x^2 - 2cx$. Add and subtract $c^2$:

Let $R^2 = c^2 + 2C$ (which is a constant, since $c$ is fixed and $C$ is arbitrary). Then:

$(x - c)^2 + y^2 = R^2$ …

Method: Naming the Family by Integrating the DE

To decide what geometric family a differential equation represents, integrate it to an algebraic relation and rewrite that relation into a recognisable standard form.

Steps

Step 1: Rearrange into integrable differentials.

$y\frac{dy}{dx}+x=c$ becomes $y\,dy+x\,dx=c\,dx$.

Step 2: Integrate both sides.

Step 3: Complete the square to identify the curve. …

Common Mistakes

Mistake 1: Naming the family without integrating first.

Why it's wrong: the shape only becomes clear after integrating $y\frac{dy}{dx}+x=c$ to $\frac{x^2}{2}+\frac{y^2}{2}=cx+k$. Correct approach: integrate, then rewrite into standard form to identify the curve.

Mistake 2: Misreading $x^2+y^2=2cx+2k$ as an ellipse or parabola. …