Comprehensive Notes on Differential Equations

Fundamentals and Definition of Differential Equations

• An equation relating a dependent variable to one or more independent variables by means of its differential coefficients (derivatives) with respect to those independent variables is defined as a differential equation.

• The variable being differentiated is the dependent variable ($y$), which depends on one or more independent variables ($x$).

• Differential equations are widely used to model real-world physical phenomena across engineering, physics, and biology:

  • Population Dynamics: \n\frac{dP}{dt} = kP\n

  • Falling Body under Gravity: \n\frac{d^2x}{dt^2} = -g\n

  • Electric Circuit Current Flow: \nL \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = E(t)\n

  • Wave Equation: \n\frac{\partial^2 u}{\partial x^2} = c^2 \frac{\partial^2 u}{\partial t^2}\n

  • Heat Conduction / Energy Equation: \n\rho C_p \frac{\partial T}{\partial \theta} = k \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right)\n

Classification of Differential Equations

• Differential equations are divided into two main categories based on the number of independent variables:

• Ordinary Differential Equations (ODE):

  • Differential equations involving only one independent variable and ordinary derivatives of the dependent variable with respect to that single independent variable.

  • Example 1: \n\left(\frac{dy}{dx}\right)^2 - 3 \frac{dy}{dx} + 2y = e^x\n

  • Example 2: \n\frac{d^2y}{dx^2} - 5 \frac{dy}{dx} + 3y = 0\n

  • Example 3: \n\left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^{\frac{1}{2}} = k \frac{d^2y}{dx^2}\n

  • Example 4: \n\frac{d^4y}{dx^4} + 4 \frac{dy}{dx} = e^x \cos(x)\n

• Partial Differential Equations (PDE):

  • Differential equations involving more than one independent variable (e.g., $x, y, z, \theta$) and partial derivatives of the dependent variable with respect to those independent variables.

  • Example 1: \nx \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0\n

  • Example 2: \n\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0\n

  • Example 3: \n\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = x + y\n

Standard Notations for Derivatives

• Leibniz Notation: \n\frac{dy}{dx}, \frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}, \dots, \frac{d^n y}{dx^n}\n

  • Explicitly displays both the dependent variable and the independent variable.

• Prime Notation: \ny', y'', y''', \dots, y^{(n)}\n

  • Represents sequential derivatives with respect to the primary independent variable.

• Newton's Dot Notation: \n\dot{s}, \ddot{s}, \dddot{s}\n

  • Exclusively used to denote derivatives taken with respect to time ($t$).

• Subscript Notation: \nu_{xx}, u_{tt}\n

  • Used to concisely represent partial derivatives with respect to specified independent variables.

Order and Degree of Differential Equations

• Order:

  • The order of a differential equation is the order of the highest-order derivative present in the equation.

  • For example, in the differential equation: \nx^2 \left(\frac{d^2y}{dx^2}\right)^3 + 3 \left(\frac{d^3y}{dx^3}\right)^2 + 7 \frac{dy}{dx} - 4y = 0\n

  • The derivatives present have orders 3 ($\frac{d^3y}{dx^3}$), 2 ($\frac{d^2y}{dx^2}$), and 1 ($\frac{dy}{dx}$). The highest order is 3; therefore, the order of the equation is 3.

• Degree:

  • The degree of a differential equation is the power/exponent of the derivative of the highest order present in the equation, provided the equation is written in a rationalized form free from radicals and fractional exponents regarding the differential coefficients.

  • In the example above, the highest-order derivative is $\frac{d^3y}{dx^3}$, which is raised to the power of 2. Thus, the degree of the equation is 2.

• Examples of Order and Degree Analysis:

  • Equation: \n\frac{dy}{dx} = \sin(x)\n

  • Order: 1 (One)

  • Degree: 1 (One)

  • Equation: \n\left(\frac{dy}{dx}\right)^2 + 3y^2 = 5x\n

  • Order: 1 (One)

  • Degree: 2 (Two)

  • Equation: \n\left(\frac{d^2s}{dt^2}\right)^2 + t^2 \left(\frac{ds}{dt}\right)^4 = 0\n

  • Order: 2 (Two)

  • Degree: 2 (Two)

  • Equation: \n\frac{d^3v}{dr^3} + \frac{2}{r} \frac{dv}{dr} = 0\n

  • Order: 3 (Three)

  • Degree: 1 (One)

  • Equation: \n\left(\frac{d^4s}{dx^4}\right)^2 + x^3 \left(\frac{d^3y}{dx^3}\right)^5 = \sin(x)\n

  • Order: 4 (Four)

  • Degree: 2 (Two)

Classification by Linearity

• Criteria for a Linear Differential Equation:

  1. The dependent variable ($y$) and all of its derivatives ($\frac{dy}{dx}, \frac{d^2y}{dx^2}, \dots$) are of first degree (i.e., the exponent of each term involving $y$ or its derivatives is 1).

  2. The coefficient of each term depends at most on the independent variable ($x$).

  3. Neither the dependent variable nor any of its derivatives is inside a non-linear function (such as trigonometric, exponential, or logarithmic functions).

• Nonlinear Differential Equations:

  • Any differential equation that fails one or more of the three linearity criteria above.

• Detailed Examples:

  • \n\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} + 9y = 0\n

  • Linear: All derivatives and dependent variables are of degree 1, with constants/independent variable coefficients.

  • \n\frac{d^3y}{dx^3} + \left(\frac{dy}{dx}\right)^4 + 6y = 10\n

  • Nonlinear: The derivative term $\frac{dy}{dx}$ is raised to the 4th power (degree 4 instead of 1).

  • \nx^2 \frac{d^2y}{dx^2} + y \frac{dy}{dx} = x^3\n

  • Nonlinear: The term $y \frac{dy}{dx}$ contains the product of the dependent variable $y$ and its derivative $\frac{dy}{dx}$.

  • \n\frac{dy}{dx} = \sin(y)\n

  • Nonlinear: The dependent variable $y$ appears as the argument of the non-linear sine function.

  • \nx^3 \frac{d^3y}{dx^3} - x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + 5y = e^x\n

  • Linear: All derivatives and dependent variables are of power 1, and coefficients depend solely on the independent variable $x$.

Solutions of Differential Equations

• General Solution (Complete Primitive):

  • An equation or algebraic relation containing $n$ arbitrary independent constants that satisfies an ordinary differential equation of the $n$th order.

• Particular Solution:

  • A solution obtained directly from the general solution by assigning specific numerical values to the arbitrary constants.

Elimination of Arbitrary Constants

• Fundamental Rules for Elimination:

  • The number of times differentiation must be performed is equal to the number of arbitrary constants to be eliminated.

  • The order of the resulting differential equation is equal to the total number of arbitrary constants eliminated.

  • The final differential equation must be completely free from all arbitrary constants and consistent with the original equation.

• Example 1: Eliminate arbitrary constants $a$ and $b$ from y = a e^{3x} + b e^x$.\n - Given equation: y = a e^{3x} + b e^x \quad \text{--- (1)} \n - First differentiation: \frac{dy}{dx} = 3a e^{3x} + b e^x \quad \text{--- (2)} \n - Second differentiation: \frac{d^2y}{dx^2} = 9a e^{3x} + b e^x \quad \text{--- (3)} \n - Subtract equation (1) from (2): \frac{dy}{dx} - y = 2a e^{3x} \quad \text{--- (4)} \n - Subtract equation (2) from (3): \frac{d^2y}{dx^2} - \frac{dy}{dx} = 6a e^{3x} = 3\left(2a e^{3x}\right) \n - Substitute (4) into the right side: \frac{d^2y}{dx^2} - \frac{dy}{dx} = 3\left(\frac{dy}{dx} - y\right) \n - Rearranging yields the differential equation: \frac{d^2y}{dx^2} - 4 \frac{dy}{dx} + 3y = 0 \n\n- **Example 2:** Eliminate arbitrary constants a$and$b$from$y = a \tan(x) + b \sec(x).\n - Multiply both sides by \cos(x)$:$ y \cos(x) = a \sin(x) + b \n - Differentiate with respect to x$using the product rule:$ y (-\sin(x)) + \frac{dy}{dx} \cos(x) = a \cos(x) \n - Divide through by \cos(x)$:$ -y \tan(x) + \frac{dy}{dx} = a \quad \text{--- (1)} \n - Differentiate equation (1) with respect to x$:$ \frac{d^2y}{dx^2} - \left( y \sec^2(x) + \frac{dy}{dx} \tan(x) \right) = 0 \n - Simplification gives: \frac{d^2y}{dx^2} - \frac{dy}{dx} \tan(x) - y \sec^2(x) = 0 \n\n- **Example 3:** Eliminate c_1$and$c_2$from$x = c_1 \cos(\omega t) + c_2 \sin(\omega t)$where$\omega is a fixed parameter.\n - Given equation: x = c_1 \cos(\omega t) + c_2 \sin(\omega t) \quad \text{--- (1)} \n - First derivative with respect to t$:$ \frac{dx}{dt} = -\omega c_1 \sin(\omega t) + \omega c_2 \cos(\omega t) \n - Second derivative with respect to t$:$ \frac{d^2x}{dt^2} = -\omega^2 c_1 \cos(\omega t) - \omega^2 c_2 \sin(\omega t) = -\omega^2 \left( c_1 \cos(\omega t) + c_2 \sin(\omega t) \right) \n - Substitute equation (1) into the expression: \frac{d^2x}{dt^2} = -\omega^2 x \n - Final differential equation: \frac{d^2x}{dt^2} + \omega^2 x = 0 \n\n# Family of Curves and Geometric Applications\n\n- **Family of Curves Concept:**\n - A family of curves on a plane is defined by an equation containing one or more parameters along with spatial coordinates. The differential equation representing the family is established by eliminating the parameters.\n\n- **Standard Algebraic Formulas for Curves:**\n - **Equations of a Straight Line:**\n - General Form: Ax + By + C = 0\n - Point-Slope Form: y - y_1 = m(x - x_1)\n - Slope-Intercept Form: y = mx + b\n - Intercept Form: \frac{x}{a} + \frac{y}{b} = 1\n - Normal Form: x \cos(\alpha) + y \sin(\alpha) = \rho\n - **Circle** (center (h, k)$, radius$r$):$ (x - h)^2 + (y - k)^2 = r^2 \n - **Ellipse** (center (h, k)$, semi-axes$a$and$b$):$ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \quad \text{or} \quad \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \n - **Hyperbola** (center (h, k)$, semi-transverse axis$a$, semi-conjugate axis$b$):$ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \quad \text{or} \quad \frac{(x - h)^2}{b^2} - \frac{(y - k)^2}{a^2} = 1 \n - **Parabola** (vertex (h, k)$, focal distance$a$):$ (y - k)^2 = 4a(x - h) \quad \text{or} \quad (x - h)^2 = 4a(y - k) \n - **Equilateral Hyperbola** (center at origin): xy = c \n\n- **General Steps for Geometric Problems:**\n 1. Draw/visualize the figure corresponding to the given geometric condition.\n 2. Formulate the algebraic equation from geometric properties.\n 3. Identify all arbitrary parameters/constants.\n 4. Eliminate the arbitrary parameters by differentiation and simplify.\n\n# Worked Examples of Geometric Families of Curves\n\n- **Example 1:** Find the differential equation of all circles of fixed radius a$and center$(h, k).\n - Circle equation: (x - h)^2 + (y - k)^2 = a^2 \quad \text{--- (1)} \n - Differentiate twice with respect to x$to eliminate arbitrary parameters$h$and$k:\n - First derivative: 2(x - h) + 2(y - k)y' = 0 \implies (x - h) + (y - k)y' = 0 \quad \text{--- (2)} \n - Second derivative: 1 + (y')^2 + (y - k)y'' = 0 \quad \text{--- (3)} \n - Express (y - k)$and$(x - h) from (3) and (2):\n - From (3): y - k = -\frac{1 + (y')^2}{y''} \n - From (2): x - h = -(y - k)y' = \frac{1 + (y')^2}{y''} y' \n - Substitute these expressions into circle equation (1): \left( \frac{1 + (y')^2}{y''} y' \right)^2 + \left( -\frac{1 + (y')^2}{y''} \right)^2 = a^2 \n - Simplify by factoring \frac{(1 + (y')^2)^2}{(y'')^2}$:$ \frac{(1 + (y')^2)^2}{(y'')^2} \left( (y')^2 + 1 \right) = a^2 \n - Resulting differential equation: \left( 1 + (y')^2 \right)^3 = a^2 (y'')^2 \n\n- **Example 2:** Find the differential equation of all circles passing through the origin with centers on the x-axis.\n - Since the center lies on the x-axis, its coordinates are (a, 0)$. Because each circle passes through the origin, its radius is$a\n - Circle equation: (x - a)^2 + y^2 = a^2 \implies x^2 - 2ax + a^2 + y^2 = a^2 \implies x^2 + y^2 = 2ax \quad \text{--- (1)} \n - Differentiate equation (1) with respect to x$:$ 2x + 2y \frac{dy}{dx} = 2a \implies a = x + y \frac{dy}{dx} \n - Substitute a$back into$(x - a)^2 + y^2 = a^2$:$ \left( x - \left( x + y \frac{dy}{dx} \right) \right)^2 + y^2 = \left( x + y \frac{dy}{dx} \right)^2 \n - Simplify both sides: \left( -y \frac{dy}{dx} \right)^2 + y^2 = x^2 + 2xy \frac{dy}{dx} + \left( y \frac{dy}{dx} \right)^2 \n - Subtract \left( y \frac{dy}{dx} \right)^2$from both sides to obtain:$ y^2 = x^2 + 2xy \frac{dy}{dx} \n\n- **Example 3:** Form the differential equation representing the family of curves y = a \cos(x + b).\n - Given: y = a \cos(x + b) \quad \text{--- (1)} \n - First derivative: \frac{dy}{dx} = -a \sin(x + b) \quad \text{--- (2)} \n - Second derivative: \frac{d^2y}{dx^2} = -a \cos(x + b) \quad \text{--- (3)} \n - Comparing equations (1) and (3):\n     \frac{d^2y}{dx^2} = -y \implies \frac{d^2y}{dx^2} + y = 0 \n\n- **Example 4:** Find the differential equation of the family of parabolas having vertices at the origin and foci on the y-axis.\n  \n  \n\n - Equation of this family: y = ax^2 \quad (a eq 0) \n - Differentiate with respect to x$:$ y' = 2ax \implies a = \frac{y'}{2x} \n - Substitute a$into the original equation:$ y = \left( \frac{y'}{2x} \right) x^2 = \frac{x y'}{2} \n - Final differential equation: xy' - 2y = 0 \n - Note that this is a **first-order linear differential equation**.\n\n- **Example 5:** Find the differential equation of the family of circles having their centers on the y-axis.\n  \n  \n\n - A member of this family has its center anywhere on the y-axis (0, b)$and radius$r$of arbitrary magnitude (2-parameter family: parameters$b$and$r).\n - Equation: x^2 + (y - b)^2 = r^2 \n - First differentiation with respect to x$:$ 2x + 2(y - b)y' = 0 \implies x + (y - b)y' = 0 \implies y - b = -\frac{x}{y'} \n - Re-arranging gives parameter b$:$ b = \frac{x + yy'}{y'} \n - Differentiate x + (y - b)y' = 0$a second time using the product rule:$ 1 + y' y' + (y - b)y'' = 0 \implies 1 + (y')^2 + (y - b)y'' = 0 \n - Substitute y - b = -\frac{x}{y'}$into the second-derivative equation:$ 1 + (y')^2 + \left( -\frac{x}{y'} \right) y'' = 0 \n - Multiply the entire equation by y'$:$ y' + (y')^3 - xy'' = 0 \n - Rearranging yields the final second-order differential equation: xy'' - (y')^3 - y' = 0 $$