Chapter 9 Differential Equations

Notes

Differential Equations – Complete Notes

1. Differential Equation

An equation involving an independent variable, a dependent variable and derivatives of the dependent variable with respect to the independent variable is called a differential equation.

Examples:

• dy/dx = 2x
• d²y/dx² + 3dy/dx + y = 0
• (dy/dx)² + y = x

1.1 Derivatives Used in Differential Equations

• First derivative: dy/dx or y'
• Second derivative: d²y/dx² or y''
• Third derivative: d³y/dx³ or y'''
• Fourth derivative: d⁴y/dx⁴ or y''''

2. Order of a Differential Equation

The order of a differential equation is the order of the highest-order derivative present in the equation.

Examples:

• dy/dx + y = 0 → Order = 1
• d²y/dx² + dy/dx + y = 0 → Order = 2
• d³y/dx³ + y = 0 → Order = 3

3. Degree of a Differential Equation

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

The degree is defined only when the differential equation can be expressed in polynomial form in derivatives.

3.1 Important Points

• Order is determined by the highest-order derivative.
• Degree is determined by the power of the highest-order derivative in polynomial form.
• The degree may not be defined when derivatives occur inside trigonometric, logarithmic or exponential functions and the equation cannot be converted into polynomial form.

3.2 Examples on Order and Degree

Example 1:

d⁴y/dx⁴ + sin(y''') = 0

Order = 4

Degree = Not defined, because a derivative occurs inside a trigonometric function.

Example 2:

y'' + 5 = 0

Order = 2

Degree = 1

Example 3:

(d⁵y/dx⁵)⁴ + 3(d²y/dx²) = 0

Order = 5

Degree = 4

Example 4:

d²y/dx² + cos(dy/dx) = 0

Order = 2

Degree = Not defined.

Example 5:

(y''')² + (y'')³ + (y')⁴ + 15 = 0

Order = 3

Degree = 2

Example 6:

(y''')³ + (y'')² + sin(y') + 1 = 0

Order = 3

Degree = Not defined.

4. Solution of a Differential Equation

A solution of a differential equation is a function that satisfies the given differential equation.

4.1 General Solution

A solution containing arbitrary constants is called a general solution.

The number of independent arbitrary constants in the general solution is generally equal to the order of the differential equation.

Example:

y = mx + c

Here, m and c are two arbitrary constants.