Q.Find the equation of the curve passing through the point $(1, 1)$ whose differential equation is $x\,dy = (2x^2 + 1)\,dx$ $(x \neq 0)$.

Concept: Separation of Variables — the equation is already in separated form, so we integrate directly.

Step 1 – Separate and integrate

The given equation is

Divide through by $x$ (allowed since $x \neq 0$):

Integrate both sides:

Step 2 – Perform the integration

Step 3 – Apply the initial condition

The curve passes through $(1, 1)$: …

The key idea here is separation of variables. The differential equation is given in a form where $dy$ and $dx$ are already on opposite sides of the equals sign — but not quite separated, because $x$ multiplies $dy$. That’s a small fix: just divide both sides by $x$ (and $x \neq 0$ is given, so we’re safe). Once $dy$ is alone, the right-hand side becomes a function of $x$ only, and we can integrate both sides directly.

Why does this work? Because the equation is of the form $\frac{dy}{dx} = f(x)$ — there’s no $y$ on the right. That’s the simplest possible case of separation: you don’t even need to rearrange terms involving $y$. Just integrate.

Let’s go step by step.

1. Rewrite the equation in separated form We have $x\,dy = (2x^2 + 1)\,dx$. Divide through by $x$ (allowed since $x \neq 0$):

Simplify the fraction:

So the equation becomes

2. Integrate both sides The left side integrates to $y$ (plus a constant). The right side is a sum of simple terms:

This gives

where $C$ is the constant of integration.

3. Use the given point to find $C$ …

Method: Direct integration (simplest separable case) plus a point condition

Use this when the equation reduces to $dy = F(x)\,dx$ — there is no $y$ on the right at all — and a point is given so the curve is determined uniquely.

Steps

Step 1: Isolate $dy$ and simplify the right side to something integrable.

Divide through so that $dy$ stands alone, then simplify the $x$-expression. A quotient like $\dfrac{2x^2+1}{x}$ should be split into separate power terms, e.g. $2x + \dfrac{1}{x}$, before integrating.

Step 2: Integrate directly.

$y = \int F(x)\,dx + C.$ …

Common Mistakes

Mistake 1: Not simplifying $\dfrac{2x^2+1}{x}$ before integrating.

Why it's wrong: the quotient must be split as $2x + \dfrac{1}{x}$; integrating it whole leads to errors. Correct approach: divide each term by $x$ first, then integrate to $x^2 + \log|x|$.

Mistake 2: Writing $\log x$ instead of $\log|x|$.

Why it's wrong: the general solution should be valid wherever $x \neq 0$, including negative $x$; dropping the absolute value narrows the domain. Correct approach: keep $\log|x|$. …