Class 12 Maths Differential Equations Notes

9.1 Introduction

A differential equation is an equation that relates a function with its derivatives.

• It represents the rate of change of a quantity
• Widely used in physics, biology, engineering, and economics
• Helps in modeling real-life phenomena like population growth, motion, and heat flow

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9.2 Basic Concepts

• Order of a Differential Equation: Highest derivative present in the equation
• Degree of a Differential Equation: Power of the highest derivative after removing radicals and fractions
• Solution of a Differential Equation: A function that satisfies the differential equation

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9.3 General and Particular Solutions of a Differential Equation

🔹 General Solution

• Contains all possible solutions of the differential equation
• Involves arbitrary constants equal to the order of the equation

Example:$$\frac{dy}{dx} = 3x^2 \quad \Rightarrow \quad y = x^3 + C$$dxdy​=3×2⇒y=x3+C

🔹 Particular Solution

• Obtained by assigning values to arbitrary constants using initial or boundary conditions

Example: If y = x³ + C and y = 2 when x = 1, then$$2 = 1 + C \Rightarrow C = 1$$2=1+C⇒C=1

So the particular solution is $y = x^3 + 1$y=x3+1

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9.4 Methods of Solving First Order, First Degree Differential Equations

🔹 1. Variable Separable Method

• Equations of the form:

$$\frac{dy}{dx} = g(x)h(y)$$dxdy​=g(x)h(y)

• Rewrite as:

$$\frac{dy}{h(y)} = g(x) dx$$h(y)dy​=g(x)dx

• Integrate both sides to find general solution

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🔹 2. Homogeneous Equations

• Equations where both sides are homogeneous functions of the same degree
• Substitution $y = vx$y=vx simplifies the equation to separable form

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🔹 3. Linear Equations

• Standard form:

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

• Solve using integrating factor (I.F.):

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

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🔹 4. Exact Equations

• If an equation can be written as:

$$M(x,y)dx + N(x,y)dy = 0$$M(x,y)dx+N(x,y)dy=0

• And satisfies: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$∂y∂M​=∂x∂N​
• Then integrate to find solution

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✅ Key Points to Remember

• Order = highest derivative, Degree = power of highest derivative
• General solution contains arbitrary constants; particular solution satisfies given conditions
• Methods for first-order, first-degree equations include:
  • Separable
  • Homogeneous
  • Linear
  • Exact
• Differential equations model real-world phenomena efficiently