Differential Equations & Area Under Curves

Overview

A differential equation establishes a relation between an unknown function $y=f(x)$ and its derivatives $\frac{dy}{dx},\frac{d^{2}y}{d{x}^2},\dots$. In IIT-JEE Advanced, differential equations frequently emerge from geometrical tangents/normals, physics rate laws, and bounded planar areas.

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1. Linear Differential Equations (First Order)

📊Visual Mathematical Intuition

Bounded Area Between Curves & Differential Slope Field

Bounded Area:$A = \int_a^b |f(x) - g(x)| dx = \int_a^b (y_{\text{upper}} - y_{\text{lower}}) dx$

Differential Law:$\frac{dy}{dx} = f(x, y) \implies y e^{\int P dx} = \int Q e^{\int P dx} dx$

Bounded analytical area enclosed between intersecting curves $\int [f(x) - g(x)] dx$, alongside the directional slope field tangents guiding solution trajectory curves.

$$\frac{dy}{dx}+P(x)y=Q(x)$$

Integrating Factor (I.F.)

$$\text{I.F.}=e^{\int P(x)dx}$$

General Solution

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

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2. Bernoulli's Differential Equation

$$\frac{dy}{dx}+P(x)y=Q(x)y^n(n\ne 0,1)$$

Divide by $y^n$:

$$y^{-n}\frac{dy}{dx}+P(x)y^{1-n}=Q(x)$$

Substitute $v=y^{1-n}⟹\frac{dv}{dx}=(1-n)y^{-n}\frac{dy}{dx}$, reducing the equation to standard linear form:

$$\frac{dv}{dx}+(1-n)P(x)v=(1-n)Q(x)$$

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3. Exact Differential Forms (Instant Speed Secrets)

$$\begin{array}{rl}d(xy)& =xdy+ydx\\ d\left(\frac{x}{y}\right)& =\frac{ydx-xdy}{y^2}\\ d\left(\frac{y}{x}\right)& =\frac{xdy-ydx}{x^2}\\ d(\arctan \left(\frac{y}{x}\right))& =\frac{xdy-ydx}{x^2+y^2}\\ d(\ln (xy))& =\frac{xdy+ydx}{xy}\\ d(\sqrt{x^2+y^2})& =\frac{xdx+ydy}{\sqrt{x^2+y^2}}\end{array}$$

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4. Area Under Curves (Planar Quadrature)

The area bounded between two continuous curves $y=f(x)$ (upper) and $y=g(x)$ (lower) from $x=a$ to $x=b$:

$$\text{Area}={\int }_a^b|f(x)-g(x)|dx$$

Standard Parabolic & Elliptic Area Results

• Standard Parabola & Line: Area bounded by $y^2=4ax$ and $y=mx$:$$\text{Area}=\frac{8a^2}{3m^3}$$
• Intersection of Two Parabolas: Area bounded by $y^2=4ax$ and $x^2=4by$:$$\text{Area}=\frac{16ab}{3}$$
• Ellipse: Area of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$:$$\text{Area}=\pi ab$$