Class 12 Notes - Differential Equations

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1. Introduction to Differential Equations

A differential equation is an equation that involves derivatives of a function. It relates a function with its derivatives and is fundamental in describing various physical phenomena such as motion, growth, decay, and heat flow.

2. Order and Degree of a Differential Equation

  • Order: The order of a differential equation is the order of the highest derivative present in the equation.
  • Degree: The degree is the power of the highest order derivative, provided the equation is polynomial in derivatives.
Example: In d2y/dx2 + 3 dy/dx + 2y = 0, the order is 2 and the degree is 1.

3. Types of Differential Equations

  • Ordinary Differential Equations (ODE): Involves derivatives with respect to a single variable.
  • Partial Differential Equations (PDE): Involves partial derivatives with respect to more than one variable.

4. General and Particular Solutions

  • General Solution: Contains arbitrary constants and represents a family of solutions.
  • Particular Solution: Obtained by assigning specific values to the arbitrary constants using initial or boundary conditions.

5. Solving First Order, First Degree Differential Equations

5.1. Variable Separable Method

If the equation can be written as dy/dx = f(x)g(y), separate the variables and integrate both sides.

Example: dy/dx = x y
Separate: dy/y = x dx
Integrate: ln|y| = (1/2)x2 + C

5.2. Homogeneous Equations

An equation of the form dy/dx = F(y/x) is homogeneous. Substitute y = vx and solve.

5.3. Linear Differential Equations

Standard form: dy/dx + P(x)y = Q(x)
Use integrating factor (IF): IF = e∫P(x)dx

Example: dy/dx + y = ex
IF = e∫1 dx = ex
Solution: y·ex = ∫ex·ex dx = ∫e2x dx = (1/2)e2x + C

6. Applications of Differential Equations

  • Population growth and decay
  • Radioactive decay
  • Newton’s law of cooling
  • Simple harmonic motion
  • Electrical circuits

7. Practice Problems

  1. Solve: dy/dx = 2x
  2. Solve: dy/dx + y = 0
  3. Solve: dy/dx = (x + y) using substitution.
  4. Find the general solution of dy/dx = y tan x.
  5. Application: A tank contains 100 liters of salt water. Fresh water flows in at 5 liters/min and the mixture flows out at the same rate. Set up the differential equation for the amount of salt in the tank at time t.

8. Summary

  • Differential equations relate functions and their derivatives.
  • Order and degree are key characteristics.
  • First order equations can often be solved by separation of variables, substitution, or integrating factor.
  • Applications are found in science, engineering, and economics.