📌 1. Introduction

A differential equation is an equation involving:

• A function
• Its derivatives

👉 It shows how a quantity changes with respect to another.

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📖 Example:

$$\frac{dy}{dx} = x$$dxdy​=x

This means rate of change of $y$y with respect to $x$x is equal to $x$x.

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📌 2. Definition

A differential equation is an equation involving derivatives of a dependent variable with respect to an independent variable.

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📊 General Form:

$$\frac{dy}{dx} = f(x, y)$$dxdy​=f(x,y)

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$\frac{dy}{dx} = f(x, y)$dxdy​=f(x,y)

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📌 3. Order and Degree

🔹 Order

The highest order derivative present.

👉 Example:$$\frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0$$dx2d2y​+dxdy​+y=0

Order = 2

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🔹 Degree

Power of highest order derivative (when equation is polynomial in derivatives).

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📌 4. Types of Differential Equations

🔹 1. Ordinary Differential Equation (ODE)

Involves one independent variable

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🔹 2. Partial Differential Equation (PDE)

Involves partial derivatives

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📌 5. Solution of Differential Equation

A function that satisfies the differential equation.

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📊 Types of Solutions

1. General Solution

Contains arbitrary constants

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2. Particular Solution

Obtained by assigning values to constants

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📌 6. Formation of Differential Equations

Formed by eliminating constants from given family of curves.

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📌 7. Methods of Solving Differential Equations

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🔹 1. Variable Separable Method

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

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

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Steps:

• Separate variables
• Integrate both sides

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📖 Example:

$$\frac{dy}{dx} = xy$$dxdy​=xy $$\frac{dy}{y} = xdx$$ydy​=xdx

Integrate:$$\ln y = \frac{x^2}{2} + C$$lny=2×2​+C

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📌 8. Homogeneous Differential Equations

Form:$$\frac{dy}{dx} = \frac{f(x,y)}{g(x,y)}$$dxdy​=g(x,y)f(x,y)​

Where both functions are homogeneous.

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Substitution:

$$y = vx$$y=vx

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📌 9. Linear Differential Equation

Standard form:$$\frac{dy}{dx} + Py = Q$$dxdy​+Py=Q

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$\frac{dy}{dx} + Py = Q$dxdy​+Py=Q

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🔹 Integrating Factor (IF)

$$IF = e^{\int P dx}$$IF=e∫Pdx

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$IF = e^{\int P dx}$IF=e∫Pdx

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Solution:

$$y \cdot IF = \int Q \cdot IF \, dx$$y⋅IF=∫Q⋅IFdx

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📌 10. Applications of Differential Equations

• Population growth
• Radioactive decay
• Cooling law
• Motion problems
• Economics

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📌 11. Important Results

1. Exponential Growth

$$y = Ce^{kx}$$y=Cekx

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$y = Ce^{kx}$y=Cekx

$c$c

$k$k

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2. Decay Law

$$y = Ce^{-kx}$$y=Ce−kx

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📌 12. Graphical Interpretation

• Solution curves represent family of curves
• Each solution corresponds to different constant

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❗ Common Mistakes

• Not separating variables properly
• Missing constant of integration
• Wrong integrating factor
• Algebraic mistakes

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🧠 Exam Tips

• Practice solving methods
• Learn formulas
• Focus on steps
• Solve NCERT examples

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📚 Practice Questions

1. Solve differential equation
2. Find general solution
3. Form equation from family
4. Apply linear method

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🎯 Conclusion

Differential Equations help us model real-life situations and understand how variables change. This chapter is essential for higher studies in mathematics, physics, and engineering.

Mastering solving techniques ensures strong performance in exams.