Chapter 11
Linear Differential Equations of Second
and Higher Order
11.1 Introduction
A differential equation of the form =0 in which the
dependent variable and its derivatives viz. , etc occur in first
degree and are not multiplied together is called a Linear Differential Equation.
11.2 Linear Differential Equations (LDE) with Constant Coefficients
A general linear differential equation of nth order with constant coefficients is
given by:
where are constant and is a function of alone or constant.
Or , where , , ….., are called
differential operators.
11.3 Solving Linear Differential Equations with Constant Coefficients
Complete solution of equation is given by C.F + P.I.
where C.F. denotes complimentary function and P.I. is particular integral.
When , then solution of equation is given by C.F
11.3.1 Rules for Finding Complimentary Function (C.F.)
Consider the equation
Step 1: Put , auxiliary equation (A.E) is given by
……③
Step 2: Solve the auxiliary equation given by ③
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, I. If the n roots of A.E. are real and distinct say , ,…
C.F. =
II. If two or more roots are equal i.e. = =… ,
C.F. =
III. If A.E. has a pair of imaginary roots i.e. ,
C.F. =
IV. If 2 pairs of imaginary roots are equal i.e. ,
C.F. =
Example 1 Solve the differential equation:
Solution:
Auxiliary equation is:
C.F. =
Since solution is given by C.F
Example 2 Solve the differential equation:
Solution:
Auxiliary equation is: …….①
By hit and trial is a factor of ①
∴① May be rewritten as
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,C.F. =
Since solution is given by C.F
Example 3 Solve
Solution: Auxiliary equation is:
…….①
By hit and trial is a factor of ①
∴① May be rewritten as
……②
By hit and trial is a factor of ②
∴② May be rewritten as
C.F. =
Since solution is given by C.F
Example 4 Solve the differential equation:
Solution:
Auxiliary equation is:
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, C.F. =
Since solution is given by C.F
Example 5 Solve the differential equation:
Solution:
Auxiliary equation is:
C.F. =
Since solution is given by C.F
Example 6 Solve the differential equation:
Solution:
Auxiliary equation is: …….①
By hit and trial is a factor of ①
∴① May be rewritten as
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Linear Differential Equations of Second
and Higher Order
11.1 Introduction
A differential equation of the form =0 in which the
dependent variable and its derivatives viz. , etc occur in first
degree and are not multiplied together is called a Linear Differential Equation.
11.2 Linear Differential Equations (LDE) with Constant Coefficients
A general linear differential equation of nth order with constant coefficients is
given by:
where are constant and is a function of alone or constant.
Or , where , , ….., are called
differential operators.
11.3 Solving Linear Differential Equations with Constant Coefficients
Complete solution of equation is given by C.F + P.I.
where C.F. denotes complimentary function and P.I. is particular integral.
When , then solution of equation is given by C.F
11.3.1 Rules for Finding Complimentary Function (C.F.)
Consider the equation
Step 1: Put , auxiliary equation (A.E) is given by
……③
Step 2: Solve the auxiliary equation given by ③
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, I. If the n roots of A.E. are real and distinct say , ,…
C.F. =
II. If two or more roots are equal i.e. = =… ,
C.F. =
III. If A.E. has a pair of imaginary roots i.e. ,
C.F. =
IV. If 2 pairs of imaginary roots are equal i.e. ,
C.F. =
Example 1 Solve the differential equation:
Solution:
Auxiliary equation is:
C.F. =
Since solution is given by C.F
Example 2 Solve the differential equation:
Solution:
Auxiliary equation is: …….①
By hit and trial is a factor of ①
∴① May be rewritten as
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,C.F. =
Since solution is given by C.F
Example 3 Solve
Solution: Auxiliary equation is:
…….①
By hit and trial is a factor of ①
∴① May be rewritten as
……②
By hit and trial is a factor of ②
∴② May be rewritten as
C.F. =
Since solution is given by C.F
Example 4 Solve the differential equation:
Solution:
Auxiliary equation is:
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, C.F. =
Since solution is given by C.F
Example 5 Solve the differential equation:
Solution:
Auxiliary equation is:
C.F. =
Since solution is given by C.F
Example 6 Solve the differential equation:
Solution:
Auxiliary equation is: …….①
By hit and trial is a factor of ①
∴① May be rewritten as
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