Mathematics Class 12 - Differential Equations Notes

Differential Equations

In this chapter, you will learn about differential equations, their types, methods to solve them, and their applications. By the end of this chapter, you will be able to form, solve, and interpret differential equations in various contexts.

Key Concepts

• Differential Equation: An equation involving derivatives of a function.
• Order: The order of the highest derivative in the equation.
• Degree: The power of the highest order derivative after making the equation polynomial in derivatives.
• General Solution: The solution containing arbitrary constants.
• Particular Solution: The solution obtained by giving specific values to the constants.

Types of Differential Equations

• First order, first degree (e.g., dy/dx + y = 0)
• Homogeneous and non-homogeneous equations
• Linear and non-linear equations

Formation of Differential Equations

Differential equations can be formed by eliminating arbitrary constants from a given relation.

• Differentiate the given equation as many times as there are arbitrary constants.
• Eliminate the constants to get the required differential equation.

Solving First Order, First Degree Differential Equations

• Variables separable: Write as f(y)dy = g(x)dx and integrate both sides.
• Homogeneous equations: Use substitution y = vx or x = vy.
• Linear equations: Standard form dy/dx + Py = Q; use integrating factor (IF).

General Solution and Particular Solution

• The general solution contains arbitrary constants (e.g., y = Ce^x).
• The particular solution is found by applying initial/boundary conditions.

Applications of Differential Equations

• Population growth and decay
• Radioactive decay
• Newton’s law of cooling
• Simple electric circuits (L-R, L-C circuits)

Practice Questions

1. Form the differential equation by eliminating the arbitrary constant from y = Ae^2x.
2. Solve: dy/dx = x + y.
3. Find the general solution of dy/dx + y = e^x.
4. If dy/dx = ky and y = 2 when x = 0, find the particular solution.
5. State one real-life application of differential equations.

Challenge Yourself

• Solve the homogeneous equation: (x + y)dy = (x - y)dx.
• Form the differential equation for the family of circles with center at the origin.

Did You Know?

• Differential equations are used in physics, engineering, biology, economics, and many other fields.
• Isaac Newton and Gottfried Leibniz developed calculus and the theory of differential equations.

Glossary

• Order: The highest derivative present in the equation.
• Degree: The power of the highest order derivative (when the equation is polynomial in derivatives).
• Integrating Factor (IF): A function used to solve linear differential equations.

Answers to Practice Questions

1. Differentiate: y = Ae^2x ⇒ dy/dx = 2Ae^2x = 2y ⇒ dy/dx = 2y
2. Integrating factor method or substitution: General solution is y = Ce^x - x - 1
3. IF = e^x; Solution: y = e^x(x + C)
4. General solution: y = Ce^kx; Using y = 2 at x = 0, C = 2; So, y = 2e^kx
5. Population growth, radioactive decay, Newton’s law of cooling, etc.

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Practice solving differential equations to master calculus and understand real-world changes!