Differential Equations PDF Notes, Important Questions and Synopsis
- An equation involving derivatives of a dependent variable with respect to an independent variable is called a differential equation.
The order of a differential equation is the number of the highest order derivative occurring in the differential equation.
Example: Order of the differential equation
The degree of a differential equation is the highest power (exponent) of the highest order derivative in it when it is written as a polynomial in differential coefficients.
Degree of equation
The order and the degree of a differential equation are positive integers.
A differential equation is a linear differential equation if it is expressed in the form:
A differential equation will be a non-linear differential equation if
- Its degree is more than one.
- The exponent of any differential equation is more than one.
- The exponent of the dependent variable is more than one.
- The products containing the dependent variable and its differential coefficients are present.
A function which satisfies a given differential equation is called its solution.
To formulate a differential equation:
- Consider an equation involving an independent variable, dependent variable and constants.
- Determine the number of constants and assume there be 'n' arbitrary constants.
- Differentiate the relation n times with respect to x, where n is number of arbitrary constants.
The solution which contains as many arbitrary constants as the order of the differential equation is called a general solution.
The solution which is free from arbitrary constants is called a particular solution.
The order of a differential equation is equal to the number of arbitrary constants present in the general solution.
An nth order differential equation represents an n-parameter family of curves.
Three methods of solving first-order and first-degree differential equations:
- Separating the variables if the variables can be separable.
- Substitution if the equation is homogeneous.
- Using an integrating factor if the equation is a linear differential equation.
The variable separable method is used to solve equations in which variables can be separated, i.e. the terms containing y should remain with dy and the terms containing x should remain with dx.
The solution of the differential equation , which is in the variable separable form, is given by
A differential equation of the form can be reduced to the variable separable form by substituting ax + by + c = v.
A differential equation which can be expressed in the form = f(x, y) or = g(x, y), where f(x, y) and g(x, y) are homogeneous functions, is called a homogeneous differential equation.
The degree of each term is the same in a homogeneous differential equation.
Homogeneous equations can be reduced to the variable separable form by the substitution of y = vx or x = vy.
Steps to solve a homogeneous differential equation:
Substitute y = v.x … (2)
Differentiate (2) with respect to x
Substitute and separate the variables
A differential equation of the form , where P and Q are constants or functions of x only, is called a first-order linear differential equation.
A differential equation of the form , where P1 and Q1 are constants or functions of y only, is called a first-order linear differential equation.
Solution of , where P and Q are constants or functions of x:
Integrating factor (IF) = e ∫Pdx
Solution: y (IF) = ∫ (Q × IF)dx + C
Solution of , where P1 and Q1 are constants or functions of y:
Integrating factor (IF) = e ∫P1dy
Solution: x (IF) = ∫ (Q1 × IF)dy + C
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