Differential Equations - Definition, order and degree, general and particular solutions

Definition of a Differential Equation: A differential equation is an equation that involves an independent variable, a dependent variable, and the derivatives of the dependent variable with respect to the independent variable. It acts as a mathematical rule describing how a quantity changes; for example, if we plot the solution on a graph, the differential equation defines the slope or curvature at every point $(x, y)$. Often written in the form $y' = f(x, y)$ or $F(x, y, y', y'', \dots, y^{(n)}) = 0$.

Order of a Differential Equation: The order is defined as the order of the highest order derivative appearing in the differential equation. For example, in the equation $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0$, the highest derivative is the second derivative, so the order is $2$. Visually, the order indicates the level of 'complexity' in the rate of change—first order relates to velocity/slope, while second order relates to acceleration/curvature.

Degree of a Differential Equation: The degree is the power of the highest order derivative when the differential equation is expressed as a polynomial in its derivatives. Crucially, for the degree to be defined, the equation must be a polynomial in derivatives (i.e., derivatives should not be inside functions like $\sin$, $\cos$, or $e^x$). For example, in $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 = 0$, the degree is $3$. If an equation is like $\frac{dy}{dx} + \sin\left(\frac{dy}{dx}\right) = 0$, the degree is not defined.

General Solution: A solution of a differential equation that contains as many arbitrary constants as the order of the equation is called the general solution. Geometrically, a general solution represents a 'family of curves.' For example, $y = x^2 + C$ represents an infinite set of parabolas shifted vertically, where each value of $C$ corresponds to a different curve in the family.

Particular Solution: A solution obtained from the general solution by giving specific values to the arbitrary constants is called a particular solution. These values are typically determined using given initial conditions, such as $y = y_0$ when $x = x_0$. Visually, this corresponds to selecting one specific curve from the family of curves that passes through a particular point $(x_0, y_0)$ on the Cartesian plane.

Formation of Differential Equations: To form a differential equation representing a family of curves, we differentiate the equation of the family as many times as the number of arbitrary constants present and then eliminate those constants. If a family has $n$ arbitrary constants, the resulting differential equation will be of order $n$.

Linearity of Differential Equations: A differential equation is linear if the dependent variable $y$ and its derivatives appear only in the first degree and are not multiplied together. Visually, linear equations are easier to solve and their solutions often follow the principle of superposition, where the sum of two solutions is also a solution.