Differential Equations

A differential equation (DE) is an equation containing an independent variable, a dependent variable, and the derivatives of the dependent variable with respect to the independent variable. The classification and behavior of a DE depend on its order and degree:

• Order: The order of a differential equation is the order of the highest-order derivative appearing in the equation. It is always a positive integer and is unconditionally defined for every DE.
• Degree: The degree of a differential equation is the power or exponent of the highest-order derivative, provided the equation can be expressed as a polynomial equation in terms of its derivatives. If the derivatives are embedded inside transcendental functions (such as $\sin(y')$, $\ln(y'')$, or $e^{y'}$), the degree of the equation is strictly undefined.

To form a differential equation from a given family of curves, differentiate the geometric equation successively with respect to the independent variable $x$. If the curve equation contains $n$ independent arbitrary constants, you must differentiate the equation exactly $n$ times to generate a system of equations. Eliminating the arbitrary constants from this system yields a differential equation of order $n$.

The most straightforward method for solving a first-order differential equation is the Variable Separable Form. If the equation can be factored such that the derivative $\frac{dy}{dx}$ splits into a product of a function of $x$ and a function of $y$, you can group all terms containing $y$ with the differential $dy$, and all terms containing $x$ with the differential $dx$: \[ \frac{dy}{dx} = f(x)g(y) \implies \frac{1}{g(y)} \, dy = f(x) \, dx \] Integrating both sides directly yields the general solution: $\int \frac{1}{g(y)} \, dy = \int f(x) \, dx + C$.

When a differential equation is not immediately separable, it can often be reduced to variable separable form using an algebraic substitution:

• Linear Argument Forms: For equations of the type $\frac{dy}{dx} = f(ax + by + c)$, substitute $v = ax + by + c$. Differentiating this substitution with respect to $x$ yields $\frac{dv}{dx} = a + b\frac{dy}{dx}$, which transforms the original equation into a separable equation in terms of $v$ and $x$.
• Homogeneous Forms: A differential equation is homogeneous if it can be written in the form $\frac{dy}{dx} = F\left(\frac{y}{x}\right)$. To solve this form, substitute $y = vx \implies \frac{dy}{dx} = v + x\frac{dv}{dx}$. This transformation simplifies the equation into a separable form in terms of $v$ and $x$: \[ v + x\frac{dv}{dx} = F(v) \implies x\frac{dv}{dx} = F(v) - v \implies \frac{1}{F(v) - v} \, dv = \frac{1}{x} \, dx \]

A first-order differential equation is linear if it can be written in the standard form: \[ \frac{dy}{dx} + P(x)y = Q(x) \] where $P(x)$ and $Q(x)$ are continuous functions of $x$. To solve this equation, multiply all terms by an Integrating Factor (IF), denoted by $\mu(x) = e^{\int P(x) \, dx}$. Multiplying by the integrating factor transforms the left side into the derivative of a product, allowing you to integrate directly: \[ \frac{d}{dx}\left[ y \cdot e^{\int P(x) \, dx} \right] = Q(x) \cdot e^{\int P(x) \, dx} \implies y \cdot \mu(x) = \int Q(x) \mu(x) \, dx + C \]

Two advanced variations of this first-order form are frequently tested in competitive exams:

• Bernoulli's Equation: A non-linear equation containing a power of the dependent variable on the right side: $\frac{dy}{dx} + P(x)y = Q(x)y^n$. To linearize this equation, divide all terms by $y^n$ and substitute $v = y^{1-n}$. This transforms the expression into a standard linear equation in terms of $v$ and $x$.
• Exact Differential Equations: An equation expressed in differential form, $M(x,y) \, dx + N(x,y) \, dy = 0$, is exact if its components satisfy the partial derivative condition: \[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \] The general solution is found by integrating $M$ with respect to $x$ (treating $y$ as a constant) and adding the integrals of the terms in $N$ that contain *only* $y$. If the exactness condition is not met, the equation can often be made exact by multiplying by a targeted integrating factor.

A linear second-order differential equation with constant coefficients has the standard format: \[ a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c y = f(x) \] The complete general solution consists of two independent parts added together: $y(x) = y_c(x) + y_p(x)$, where $y_c(x)$ is the Complementary Function (CF) and $y_p(x)$ is the Particular Integral (PI).

• Complementary Function ($y_c$): Found by setting the forcing function to zero ($f(x) = 0$) and solving the corresponding auxiliary quadratic equation $a m^2 + b m + c = 0$:
  1. Real and Distinct Roots ($m_1 \neq m_2$): $y_c = c_1 e^{m_1 x} + c_2 e^{m_2 x}$
  2. Real and Repeated Roots ($m_1 = m_2 = m$): $y_c = (c_1 + c_2 x)e^{m x}$
  3. Complex Conjugate Roots ($m = \alpha \pm i\beta$): $y_c = e^{\alpha x}(c_1 \cos \beta x + c_2 \sin \beta x)$
• Particular Integral ($y_p$): Accounts for the non-zero forcing function $f(x)$. It can be found using the method of undetermined coefficients (guessing a template solution based on the form of $f(x)$) or via the Variation of Parameters method for general functions.

This subsection covers several advanced geometric and analytical forms often featured in JEE Advanced:

• Clairaut's Equation: A first-order non-linear equation with the standard format $y = x p + f(p)$, where $p = \frac{dy}{dx}$. Its general solution is found by simply replacing the derivative term $p$ with an arbitrary constant $c$: $y = cx + f(c)$. This represents a family of straight lines. Differentiating with respect to $p$ yields the singular solution, which represents the envelope of these lines and contains no arbitrary constants.
• Orthogonal Trajectories: An orthogonal trajectory is a curve that intersects every member of a given family of curves at a right angle ($90^\circ$). To find the orthogonal family:
  1. Form the differential equation for the given family of curves: $F(x, y, \frac{dy}{dx}) = 0$.
  2. Replace the original derivative term $\frac{dy}{dx}$ with its negative reciprocal, $-\frac{dx}{dy}$, to enforce perpendicularity.
  3. Solve the resulting new differential equation to find the orthogonal curves.
• Slope Fields and Isoclines: Provide a method for visualizing solutions geometrically. An isocline for the equation $\frac{dy}{dx} = f(x,y)$ is a curve along which the slope of the solution path is constant ($f(x,y) = k$). Sketching these constant-slope line segments helps trace solution curves without solving the equation analytically.

Differential equations serve as the primary mathematical framework for modeling rate-dependent physical systems across science and engineering fields:

• Exponential Growth and Decay: Modeled by the first-order separable equation $\frac{dN}{dt} = kN$. If $k > 0$, the system experiences exponential growth (e.g., unconstrained population models); if $k < 0$, it experiences exponential decay (e.g., radioactive substance half-life tracking).
• Newton's Law of Cooling: States that the rate of change of the temperature $T$ of an object is directly proportional to the difference between its temperature and the surrounding ambient temperature $T_m$: \[ \frac{dT}{dt} = -k(T - T_m) \quad (\text{where } k > 0) \]
• Mixing Problems: Track the chemical salt content $S(t)$ in a fluid tank over time. The rate of change is modeled using mass balance mechanics: \[ \frac{dS}{dt} = \text{Rate in} - \text{Rate out} = (\text{Inflow Concentration} \times \text{Inflow Rate}) - \left(\frac{S(t)}{\text{Volume}} \times \text{Outflow Rate}\right) \]
• Projectile Motion with Air Resistance: Extends basic linear kinematics by adding a velocity-dependent drag force term ($F_{\text{drag}} = -cv$), transforming the system into a manageable first-order linear differential equation in terms of velocity.