Differential Equations Class 12 Notes

Essential Exam Terminology

Differential Equation

An equation that involves an independent variable, a dependent variable, and one or more derivatives of the dependent variable with respect to the independent variable.

Order

The order of the highest-order derivative appearing in the given differential equation.

Degree

The power of the highest-order derivative when the differential equation is expressed as a polynomial in its derivatives.

General Solution

The solution of a differential equation that contains as many arbitrary constants as the order of the differential equation.

Particular Solution

A solution obtained from the general solution by assigning specific values to the arbitrary constants based on given boundary conditions.

Integrating Factor (I.F.)

A function of the independent variable used to multiply a linear differential equation so that the left-hand side becomes a clean derivative of a product.

Core Concept: Order and Degree of a Differential Equation

Understanding Order and Degree is foundational for this chapter, and at least one objective question is guaranteed from this section.

1. Order: The order is always defined for every differential equation. It is simply the order of the highest derivative present (e.g., if $d^3y/dx^3$ is present, the order is 3).
2. Degree: Unlike order, the degree is not always defined. The degree is defined only when the equation can be written as a polynomial in its derivatives (i.e., terms like $y', y'', y'''$ must only have positive integer exponents and cannot be inside functions like $\sin(y')$, $e^{y'}$, or $\log(y')$). If any derivative term is trapped inside an analytical function, the degree is undefined.

Must-Remember Key Rules

• The order of a differential equation is always a positive integer and is always defined.
• The degree of a differential equation, if defined, is always a positive integer.
• The number of arbitrary constants in the general solution of a differential equation of order $n$ is exactly $n$.
• The number of arbitrary constants in any particular solution is always $0$.
• To solve variable separable equations, gather all $y$ terms with $dy$ and all $x$ terms with $dx$, then integrate both sides separately.
• A differential equation of the form $dy/dx = g(y/x)$ is homogeneous. Solve it by substituting $y = vx$, which yields $dy/dx = v + x(dv/dx)$.
• If a homogeneous equation is of the form $dx/dy = h(x/y)$, solve it by substituting $x = vy$ instead.
• For Linear Differential Equations of the form $dy/dx + Py = Q$, both $P$ and $Q$ must be either constants or functions of $x$ only.

Step-by-Step: Solving First-Order Linear Differential Equations (LDE)

1. Convert to Standard Form —
2. Identify P and Q —
3. Find the Integrating Factor (I.F.) —
4. Write and Integrate the General Solution —

Comparison: General Solution vs. Particular Solution

Worked Mini-Examples

• {"title":"Example 1: Determining Order and Degree","bodyMarkdown":"Question: Find the order and degree (if defined) of the differential equation:\n$\\left(\\frac{d^2y}{dx^2}\\right)^3 + \\left(\\frac{dy}{dx}\\right)^2 + \\sin\\left(\\frac{dy}{dx}\\right) + 1 = 0$\n\nSolution:\n- The highest derivative present is the second derivative $\\frac{d^2y}{dx^2}$. Hence, Order = 2.\n- Since the derivative term $\\frac{dy}{dx}$ is trapped inside the sine function, the equation cannot be written as a polynomial in its derivatives. Hence, the Degree is not defined."}
• {"title":"Example 2: Solving a Variable Separable Equation","bodyMarkdown":"Question: Solve the differential equation:\n$\\frac{dy}{dx} = \\frac{1 + y^2}{1 + x^2}$\n\nSolution:\n1. Separate the variables:\n$\\frac{1}{1 + y^2} \\, dy = \\frac{1}{1 + x^2} \\, dx$\n\n2. Integrate both sides:\n$\\int \\frac{1}{1 + y^2} \\, dy = \\int \\frac{1}{1 + x^2} \\, dx$\n$\\tan^{-1}(y) = \\tan^{-1}(x) + C$\nThis is the required general solution."}

Board Exam Trap Alerts

• The Constant Trap: Students frequently lose 1 mark in LDE questions by integrating the right-hand side but forgetting to add the constant of integration ($+ C$). Always add $C$ immediately after performing the main integration.
• The MCQ Trick: CBSE frequently asks: "What is the number of arbitrary constants in a particular solution of a 4th-order differential equation?" Under exam stress, students write 4. The correct answer is 0—a particular solution never contains arbitrary constants, regardless of the order!
• Homogeneous Identification: Always verify if a function is homogeneous by replacing $x$ with $\lambda x$ and $y$ with $\lambda y$. If $f(\lambda x, \lambda y) = \lambda^0 f(x, y)$, only then proceed with the $y = vx$ substitution.

Quick Revision Self-Check

• What is the degree of the differential equation: $y''' + 2(y'')^2 + y' = 0$? The highest order derivative is $y'''$ (order 3), and its exponent power is 1. Since it is a polynomial in derivatives, the degree is 1.
• State the integrating factor for the linear differential equation: $x \frac{dy}{dx} - y = 2x^2$. First, divide by $x$ to get the standard form: $\frac{dy}{dx} - \frac{1}{x}y = 2x$. Here, $P = -\frac{1}{x}$. Thus, $\text{I.F.} = e^{\int -\frac{1}{x} \, dx} = e^{-\ln(x)} = e^{\ln(1/x)} = \frac{1}{x}$.
• Which substitution is used for solving the homogeneous differential equation of the form $\frac{dx}{dy} = g\left(\frac{x}{y}\right)$? For equations of the form $\frac{dx}{dy} = g\left(\frac{x}{y}\right)$, we use the substitution $x = vy$, which implies $\frac{dx}{dy} = v + y\frac{dv}{dy}$.
• How many arbitrary constants exist in the general solution of a 3rd order differential equation? There are exactly 3 arbitrary constants, since the number of arbitrary constants in a general solution matches the order of the differential equation.

Frequently Asked Questions