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MTH 174 Lecture 819 - mth
Mathematics (Lovely Professional University)
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9 LINEAR DIFFERENTIAL EQUATIONS OF
SECOND AND HIGHER ORDER
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9.1 INTRODUCTION
dy d2 y
A differential equation in which the dependent variable, y(x) and its derivatives, say, ,
dx dx2
etc. occur in the first degree and are not multiplied together is called linear differential
equation. Then, we classify them as linear differential equation with constant co-efficients
and the other with variable coefficients.
The linear differential equations with constant coefficients generally arises in practical problems
related to the study of mechanical, acoustical and electrical vibrations, whereas linear
differential equations with variable coefficients arise generally in mathematical modeling of
physical problems. Some of the important linear differential equations with variable coefficients
are Bessel equation, Legendre’s equation, Chebyshev equation etc.
The solution of linear differential equations with constant coefficients are generally found
in terms of known standard functions while there exists no such procedure in case of
differential equations with variable coefficients and their solutions many a times results in
the form of an infinite series.
The general form of the nth order linear differential with constant coefficients is
dn y dn −1y dy
k0 n
+ k1 n −1
+ … + kn −1 + kn y = X ( x ) …(1)
dx dx dx
where k0, k1, k2,…, kn are constants and X is a function of ‘x’ only.
The general linear differential equation with variable coefficients is written as
dn y dn −1y dn − 2 y dy
P0 n + P1 n −1 + P2 n − 2 + … + Pn −1 + Pn y = X ( x) … (2)
dx dx dx dx
where P0 (≠ 0), P1, P2,…, Pn and X are function of ‘x’ only.
If X(x) = 0 in (1) and (2), then they are called linear homogeneous differential equations
with constant coefficients and variable coefficients respectively.
9.2 SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS
In equation (1), x varies on some interval of definition, say I, which may be open, semi open,
closed or infinite and the differential equation may be valid for all x ∈ (0, ∞), (–∞, 0), (–∞, ∞).
If y1(x) is a solution of the equation (1), then it must satisfy the equation identically and
577
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578 Engineering Mathematics through Applications
whence y1 (x) must be continuously differentiable (n – 1) times and dn
y1(x) must be
dxn
continuous in that interval.
Further, if coefficients P0(x), P1(x) ,…….., Pn(x), P0(x) ≠ 0, in the linear homogeneous
equation (2) are continuous on some interval of def, say I, then this equation has n linearly
independent solutions. If y1(x), y2(x) ,…, yn(x) are n linearly independent solutions, then the
general solution is their combination.
i.e. y(x) = c1y1 + c2y2 + … + cnyn … (3)
Theorem: If y1(x), y2(x) ,…, yn(x) be n linearly independent solutions of
dn y dn −1y dy
k0 n
+ k1 n −1
+ … + kn −1 + kn y = 0,
dx dx dx
where k0, k1 ,…, kn are all constants, then y = c1y1 + c2y2 + … + cnyn is also a solution of (1). This
is called the Principle of Superposition or Principle of linearity.
Proof: Putting y = c1y1 + c2y2 + … + cnyn into left hand side of equation (1), we get
dn dn −1
k0 c y + c2 y2 + … + cn yn + k1 n −1 c1 y1 + c2 y2 + … + cn yn
n 1 1
dx dx
d
+ … + kn −1 c1y1 + c2 y2 + … + cn yn + kn c1y1 + c2 y2 + … + cn yn
dx
dny dn−1y dy
= c1 k0 n1 + k1 n−11 + … + kn−1 1 + kny1
dx dx dx
dn y dn −1y dy
+ c2 k0 n2 + k1 n −12 + … + kn −1 2 + kn y2
dx dx dx
dny dn −1y dy
+…+ cn k0 nn + k1 n −1n + … + kn −1 n + kn yn
dx dx dx
= c1 [0] + c2 [0] + … + cn [0] = 0, since y1(x), y2(x), …, yn(x)
are solution of the linear equation. This proves the theorem.
Remarks: The above n linearly independent solutions y1(x), y2(x),…,yn(x) are called the fundamental solutions
of equation (1) and the set comprising them forms a basis of the nth order linear homogeneous equations.
The solution (3), y = c1y1 + c2y2 + … + cnyn = u(x) is a combination of n linearly independent
solutions containing n arbitrary constants c1, c2,…, cn. It is called the general solution of
equation (1). It is also known as the complementary function (C.F.).
Further, if y = v(x) be a solution of the non–homogeneous equation containing no arbitrary
constant is called it’s particular solution (P.I.). Therefore, y = u(x) + v(x) = C.F. + P.I. is called
the complete solution of equation (1). Hence, in order to solve equation (1), first find the
general solution (C.F.) and then find the particular solution (P.I.).
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Linear Differential Equations of Second and Higher Order 579
Linear Independence and Dependence of Solutions
Functions y1(x), y2(x) ,…, yn(x) are said to linearly independent on some interval of definition,
say I, if the relation (3) viz. c1y1 + c2y2 + … + cnyn = 0. Implies c1 = 0 = c2 = … = cn. This system
(or set) of linearly independent solutions is called fundamental system (or set) of solution
(or integrals).
However, these functions are said to be dependent on the interval of definition, say I, if
relation (3) holds for c1, c2 ,…, cn not all zero. In this case, one or more functions can be
expressed as a linear combination of the remaining functions.
1
if c1 ≠ 0, then y1 = − c2 y2 + c3 y3 + … + cn yn
c1
e.g. …(4)
Conversely, if any of yi's can be expressed as the linear combination of the remaining
functions y1, y2 ,…, yi – 1, yi + 1 ,…, yn then the given set of functions are linearly dependent.
Theorem: The necessary and sufficient condition that n integrals y1, y2, …, yn of the linear
dn y dn −1y
differential equation (2) viz. P0
n
+ P1 n −1 + … + Pn y = 0 , where P0, P1, P2 ,…, Pn (P0 ≠ 0)
dx dx
are continuous functions of x on a common interval I or constants, be linearly independent is
that the determinant, W* (Wronskian),
y1 y2 … … … … … yn
y1' y2' … … … … … yn'
… … … … … … … … does not vanish identically on I.
… … … … … … … …
n −1
y1 y2n −1 … … … … … ynn −1
Proof: Necessary Condition: If y1(x), y2(x) ,…, yn(x) are not linearly independent then there
are constants c1, c2 ,…, cn not all zero such that c1y1 + c2y2 … + cnyn = 0.
Also c1y1(i) + c2y2(i) + … + cnyn(i) = 0, i = 1, 2 ,…, n – 1.
It follows the determinant
y1 y2 … … … … … yn
y1' y2' … … … … … yn'
W * ( x) = … … … … … … … …
… … … … … … … …
y1n −1 y2n −1 … … … … … ynn −1
c1y1 + c2 y2 + ………… + cn yn y2 … … … yn
c1y1´+ c2 y2´ + ………… + cn yn´ y2 ´ … … … yn´
= …………………………… … ………
…………………………… …………
c1y1n −1 + c2 y2n −1 + … + cn ynn −1 y2n −1 … … … ynn −1
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MTH 174 Lecture 819 - mth
Mathematics (Lovely Professional University)
Scan to open on Studocu
Studocu is not sponsored or endorsed by any college or university
Downloaded by kuchy kuchy ()
, lOMoARcPSD|53991452
9 LINEAR DIFFERENTIAL EQUATIONS OF
SECOND AND HIGHER ORDER
aaaaa
9.1 INTRODUCTION
dy d2 y
A differential equation in which the dependent variable, y(x) and its derivatives, say, ,
dx dx2
etc. occur in the first degree and are not multiplied together is called linear differential
equation. Then, we classify them as linear differential equation with constant co-efficients
and the other with variable coefficients.
The linear differential equations with constant coefficients generally arises in practical problems
related to the study of mechanical, acoustical and electrical vibrations, whereas linear
differential equations with variable coefficients arise generally in mathematical modeling of
physical problems. Some of the important linear differential equations with variable coefficients
are Bessel equation, Legendre’s equation, Chebyshev equation etc.
The solution of linear differential equations with constant coefficients are generally found
in terms of known standard functions while there exists no such procedure in case of
differential equations with variable coefficients and their solutions many a times results in
the form of an infinite series.
The general form of the nth order linear differential with constant coefficients is
dn y dn −1y dy
k0 n
+ k1 n −1
+ … + kn −1 + kn y = X ( x ) …(1)
dx dx dx
where k0, k1, k2,…, kn are constants and X is a function of ‘x’ only.
The general linear differential equation with variable coefficients is written as
dn y dn −1y dn − 2 y dy
P0 n + P1 n −1 + P2 n − 2 + … + Pn −1 + Pn y = X ( x) … (2)
dx dx dx dx
where P0 (≠ 0), P1, P2,…, Pn and X are function of ‘x’ only.
If X(x) = 0 in (1) and (2), then they are called linear homogeneous differential equations
with constant coefficients and variable coefficients respectively.
9.2 SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS
In equation (1), x varies on some interval of definition, say I, which may be open, semi open,
closed or infinite and the differential equation may be valid for all x ∈ (0, ∞), (–∞, 0), (–∞, ∞).
If y1(x) is a solution of the equation (1), then it must satisfy the equation identically and
577
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578 Engineering Mathematics through Applications
whence y1 (x) must be continuously differentiable (n – 1) times and dn
y1(x) must be
dxn
continuous in that interval.
Further, if coefficients P0(x), P1(x) ,…….., Pn(x), P0(x) ≠ 0, in the linear homogeneous
equation (2) are continuous on some interval of def, say I, then this equation has n linearly
independent solutions. If y1(x), y2(x) ,…, yn(x) are n linearly independent solutions, then the
general solution is their combination.
i.e. y(x) = c1y1 + c2y2 + … + cnyn … (3)
Theorem: If y1(x), y2(x) ,…, yn(x) be n linearly independent solutions of
dn y dn −1y dy
k0 n
+ k1 n −1
+ … + kn −1 + kn y = 0,
dx dx dx
where k0, k1 ,…, kn are all constants, then y = c1y1 + c2y2 + … + cnyn is also a solution of (1). This
is called the Principle of Superposition or Principle of linearity.
Proof: Putting y = c1y1 + c2y2 + … + cnyn into left hand side of equation (1), we get
dn dn −1
k0 c y + c2 y2 + … + cn yn + k1 n −1 c1 y1 + c2 y2 + … + cn yn
n 1 1
dx dx
d
+ … + kn −1 c1y1 + c2 y2 + … + cn yn + kn c1y1 + c2 y2 + … + cn yn
dx
dny dn−1y dy
= c1 k0 n1 + k1 n−11 + … + kn−1 1 + kny1
dx dx dx
dn y dn −1y dy
+ c2 k0 n2 + k1 n −12 + … + kn −1 2 + kn y2
dx dx dx
dny dn −1y dy
+…+ cn k0 nn + k1 n −1n + … + kn −1 n + kn yn
dx dx dx
= c1 [0] + c2 [0] + … + cn [0] = 0, since y1(x), y2(x), …, yn(x)
are solution of the linear equation. This proves the theorem.
Remarks: The above n linearly independent solutions y1(x), y2(x),…,yn(x) are called the fundamental solutions
of equation (1) and the set comprising them forms a basis of the nth order linear homogeneous equations.
The solution (3), y = c1y1 + c2y2 + … + cnyn = u(x) is a combination of n linearly independent
solutions containing n arbitrary constants c1, c2,…, cn. It is called the general solution of
equation (1). It is also known as the complementary function (C.F.).
Further, if y = v(x) be a solution of the non–homogeneous equation containing no arbitrary
constant is called it’s particular solution (P.I.). Therefore, y = u(x) + v(x) = C.F. + P.I. is called
the complete solution of equation (1). Hence, in order to solve equation (1), first find the
general solution (C.F.) and then find the particular solution (P.I.).
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Linear Differential Equations of Second and Higher Order 579
Linear Independence and Dependence of Solutions
Functions y1(x), y2(x) ,…, yn(x) are said to linearly independent on some interval of definition,
say I, if the relation (3) viz. c1y1 + c2y2 + … + cnyn = 0. Implies c1 = 0 = c2 = … = cn. This system
(or set) of linearly independent solutions is called fundamental system (or set) of solution
(or integrals).
However, these functions are said to be dependent on the interval of definition, say I, if
relation (3) holds for c1, c2 ,…, cn not all zero. In this case, one or more functions can be
expressed as a linear combination of the remaining functions.
1
if c1 ≠ 0, then y1 = − c2 y2 + c3 y3 + … + cn yn
c1
e.g. …(4)
Conversely, if any of yi's can be expressed as the linear combination of the remaining
functions y1, y2 ,…, yi – 1, yi + 1 ,…, yn then the given set of functions are linearly dependent.
Theorem: The necessary and sufficient condition that n integrals y1, y2, …, yn of the linear
dn y dn −1y
differential equation (2) viz. P0
n
+ P1 n −1 + … + Pn y = 0 , where P0, P1, P2 ,…, Pn (P0 ≠ 0)
dx dx
are continuous functions of x on a common interval I or constants, be linearly independent is
that the determinant, W* (Wronskian),
y1 y2 … … … … … yn
y1' y2' … … … … … yn'
… … … … … … … … does not vanish identically on I.
… … … … … … … …
n −1
y1 y2n −1 … … … … … ynn −1
Proof: Necessary Condition: If y1(x), y2(x) ,…, yn(x) are not linearly independent then there
are constants c1, c2 ,…, cn not all zero such that c1y1 + c2y2 … + cnyn = 0.
Also c1y1(i) + c2y2(i) + … + cnyn(i) = 0, i = 1, 2 ,…, n – 1.
It follows the determinant
y1 y2 … … … … … yn
y1' y2' … … … … … yn'
W * ( x) = … … … … … … … …
… … … … … … … …
y1n −1 y2n −1 … … … … … ynn −1
c1y1 + c2 y2 + ………… + cn yn y2 … … … yn
c1y1´+ c2 y2´ + ………… + cn yn´ y2 ´ … … … yn´
= …………………………… … ………
…………………………… …………
c1y1n −1 + c2 y2n −1 + … + cn ynn −1 y2n −1 … … … ynn −1
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