Find the equation of a curve passing through origin and satisfying the differential equation (1+x²)(dy)/(dx)+2

NCERT Exemplar · Q15

(1+x^2)\frac{dy}{dx}+2xy=4x^2

Bihar BsebShort· 3mImportance★★★★★

Appeared in past exams:CBSE 2019· Set 65/1/1· 4mrewordedMHT-CET 2025· Set pcm-2025-04-25-E· 2mexact

61% · 135/222 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Integrating Factor Method

Integrating Factor Method

Some first-order differential equations refuse to separate — you cannot get all the $y$ 's on one side and all the $x$ 's on the other. The integrating factor method is the standard trick for a special (and very common) family of these: the linear first-order equation. The idea is beautifully simple: multiply the whole equation by one cleverly chosen function, and the messy left-hand side collapses into a single derivative that we can integrate directly.

The Standard Form

An equation is linear of first order if it can be written as

\frac{dy}{dx} + P\,y = Q

where $P$ and $Q$ are functions of $x$ alone (or constants). Notice $y$ and $\frac{dy}{dx}$ appear only to the first power, and never multiplied together — that is what "linear" means here.

Tip

Always rearrange into this exact shape first. The coefficient of $\frac{dy}{dx}$ must be $1$ before you read off $P$ and $Q$ .

The Integrating Factor

The magic multiplier is

\text{I.F.} = e^{\int P\,dx}.

Why this one? Multiply the equation by $e^{\int P\,dx}$ :

e^{\int P\,dx}\frac{dy}{dx} + P\,e^{\int P\,dx}\,y = Q\,e^{\int P\,dx}.

By the product rule, the left-hand side is exactly $\dfrac{d}{dx}\!\left(y\cdot e^{\int P\,dx}\right)$ , because the derivative of $e^{\int P\,dx}$ is $P\,e^{\int P\,dx}$ . So the equation becomes

\frac{d}{dx}\big(y\cdot \text{I.F.}\big) = Q\cdot \text{I.F.}

The left side is now a single derivative — that is the whole point of choosing this factor.

The Solution

Integrate both sides with respect to $x$ :

y\cdot \text{I.F.} = \int \big(Q\cdot \text{I.F.}\big)\,dx + C.

This is the general solution. In words: (solution) × (integrating factor) = integral of (Q × integrating factor), plus a constant.

A Quick Illustration

For $\dfrac{dy}{dx} + \dfrac{1}{x}\,y = x$ , we read $P = \tfrac1x$ , $Q = x$ . Then $\int P\,dx = \log x$ , so $\text{I.F.} = e^{\log x} = x$ . The solution is

y\cdot x = \int x\cdot x\,dx + C = \frac{x^3}{3} + C.

--- …

Start your 14-day free trial to unlock the full solution →

This is a first-order linear differential equation solved using the integrating factor method. The solution is $y(1+x^2) = \frac{4}{3}x^3$ , and since the curve passes through the origin, no constant term appears.

We have a differential equation that mixes $y$ and its derivative with a coefficient that depends on $x$ . The standard approach for equations of the form $\frac{dy}{dx} + P(x)y = Q(x)$ is the integrating factor method — it rewrites the left side as a single derivative, making integration straightforward.

Let’s first rewrite the given equation in that standard form.

Rewrite in standard linear form

The equation is:

(1+x^2)\frac{dy}{dx} + 2xy = 4x^2

Divide through by $(1+x^2)$ :

\frac{dy}{dx} + \frac{2x}{1+x^2}\, y = \frac{4x^2}{1+x^2}

So we identify:

P(x) = \frac{2x}{1+x^2}, \quad Q(x) = \frac{4x^2}{1+x^2}

Find the integrating factor

The integrating factor (I.F.) is:

\mu(x) = e^{\int P(x)\,dx} = e^{\int \frac{2x}{1+x^2}\,dx}

Notice that the numerator $2x$ is the derivative of the denominator $1+x^2$ . So:

\int \frac{2x}{1+x^2}\,dx = \log(1+x^2) + C

Hence:

\mu(x) = e^{\log(1+x^2)} = 1+x^2

Tip

Spotting that the numerator is the derivative of the denominator saves time — it’s a natural logarithm integral.

Multiply through by the integrating factor

Multiply the standard-form equation by $(1+x^2)$ :

(1+x^2)\frac{dy}{dx} + 2xy = 4x^2

But this is exactly the original equation! The left side is now the derivative of $y \cdot (1+x^2)$ :

\frac{d}{dx}\big[y(1+x^2)\big] = 4x^2

Note

This is the key insight: the integrating factor turns the left side into a perfect derivative. Always check that the product rule works out.

Integrate both sides …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ14 — Form the differential equation of all circles which pass through origi…→Next questionQ16 — Solve: x²(dy)/(dx)=x²+xy+y².

Q.Find the equation of a curve passing through origin and satisfying the differential equation $(1+x^2)\frac{dy}{dx}+2xy=4x^2$ .

Concept understanding — Integrating Factor Method

Integrating Factor Method

The Standard Form

The Integrating Factor

The Solution

A Quick Illustration

Method: Linear DE for a curve through a given point

Steps

Common Mistakes

Q.Find the equation of a curve passing through origin and satisfying the differential equation (1+x2)dydx+2xy=4x2(1+x^2)\frac{dy}{dx}+2xy=4x^2(1+x2)dxdy​+2xy=4x2.

🎓Unlock everything free for 14 days

Q.Find the equation of a curve passing through origin and satisfying the differential equation $(1+x^2)\frac{dy}{dx}+2xy=4x^2$ .

Unlock everything free for 14 days