Solve the simplest differential equation by direct integration.

Solve dy/dx = 2x — Simplest ODE

Question

Solve the differential equation:

\frac{dy}{dx} = 2x

Find $y$ as a function of $x$ .

Solution — Step by Step

The equation tells us that the rate of change of $y$ with respect to $x$ equals $2x$ . We need to find the function $y$ whose derivative is $2x$ . That’s a direct integration problem.

Rewrite the equation as:

dy = 2x \, dx

Now integrate both sides:

\int dy = \int 2x \, dx

The left side integrates trivially to $y$ . On the right, use $\int x^n \, dx = \frac{x^{n+1}}{n+1}$ :

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

\boxed{y = x^2 + C}

where $C$ is an arbitrary constant. This is the general solution.

Why This Works

Every differential equation of the form $\frac{dy}{dx} = f(x)$ — where the right side depends only on $x$ — can be solved by direct integration. There’s no $y$ term on the right to complicate things, so we just anti-differentiate $f(x)$ .

The constant $C$ appears because integration is the reverse of differentiation, and constants vanish when we differentiate. So infinitely many curves (a whole family of parabolas) satisfy this equation — one for each value of $C$ .

If we’re given an initial condition like $y(0) = 3$ , we substitute to get $3 = 0 + C$ , so $C = 3$ . That pins down a particular solution: $y = x^2 + 3$ .

Alternative Method

Using the antiderivative definition directly:

We know $\frac{d}{dx}(x^2) = 2x$ . So any function of the form $y = x^2 + C$ has derivative $2x$ . We can write the answer by inspection without formally integrating.

This mental approach is fast during JEE Main MCQs where you only need to verify which option satisfies the ODE. Substitute a given option back into $\frac{dy}{dx}$ and check if it equals $2x$ — takes 10 seconds.

To verify any solution to an ODE, differentiate your answer and check it matches the original equation. For $y = x^2 + C$ : $\frac{dy}{dx} = 2x$ . Matches. Done.

Common Mistake

Forgetting the constant of integration. Writing $y = x^2$ as the final answer is wrong — it’s only one particular solution, not the general solution. NCERT and CBSE board marking schemes explicitly award a mark for $+ C$ . In a 3-mark question, you lose 1 mark for this. JEE doesn’t usually penalise this way, but if the question asks for a “general solution”, dropping $C$ means your answer is incomplete.

A second slip: some students write $\int 2x \, dx = 2x^2 + C$ — they forget to divide by the new power. Remember: $\int 2x \, dx = 2 \cdot \frac{x^2}{2} + C = x^2 + C$ , not $2x^2 + C$ .

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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