DE 730 MW
Homogeneous Linear ODE with Constant Coefficients
Any homogeneous linear ordinary differential equation with constant coefficients,
dn y dn1y dy
ao n a1 n1 ... an1 an y 0
dx dx dx
may be written in the form of f(D) = 0, where f(D) is a linear differential operator.
Based on the given differential equation, an auxiliary equation is obtained, that is, f(m) = 0, where
m is any root of f(m) = 0.
Then the general solution is determined based on the type of roots obtained from the auxiliary
equation.
Nature of Roots Roots General Solution
1. Distinct roots m = a, -b
y c1eax c 2ebx
2. Equal roots m = a, a, a
y c1eax c 2 xeax c 3 x 2eax
3. Imaginary roots m = a + bi, a – bi
y c1eax cosbx c 2eax sinbx
Discriminant
is the most important part of quadratic functions when we talk about the nature of its roots. Given
a quadratic equation: y ax2 bx c , discriminant (D) can be defined mathematically as
D b2 4ac .
Discriminant Nature of Roots
D=0 Equal Roots
D>0 Real and Distinct Roots
D<0 Imaginary Roots
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DE 730 MW
Exercises
1. (D2 + 2D – 3)y = 0
2. (D2 + D – 6)y = 0
3. (D3 + 3D2 – 4D)y = 0
4. (4D3 – 7D + 3)y = 0
d3 y d2 y dy
5. 3
2 2 0
dx dx dx
6. (D2 – 6D + 9)y = 0
7. (4D3 + 4D2 + D)y = 0
8. (D4 + 6D3 + 9D2)y = 0
9. (D5 – D3)y = 0
10. (D2 – 2D + 5)y = 0
11. (D2 + 6D +13)y = 0
12. (D4 + 2D3 + 10D2)y = 0
d2 y
2
D 2D 3 y 0
dx 2
2
dy
dx
3y 0 y " 2y ' 3y 0
d d2
D D2 2
dx dx
dy d2 y
Dy D2 y 2
dx dx
D 2
2D 3 y 0 HLDE
m2 2m 3 0 auxiliary eqn
m 3 m 1 0
m 3, m 1 roots
3x
y c1e c 2e x
G.S.
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DE 730 MW
Nonhomogeneous Linear ODE with Constant Coefficients
The Method of Undetermined Coefficients
If the nonhomogeneous term d( x) in the general second‐order nonhomogeneous differential
equation
a x y" b x y' c x y d x
is of a certain special type, then the method of undetermined coefficients can be used to obtain a
particular solution
Example
Find a complete solution of the differential equation
y " 3y ' 10y 5x 2
the family of d = 5 x 2 is { x2, x, 1}; therefore, the most general linear combination of the functions
in the family is y = Ax 2 + Bx + C (where A, B, and C are the undetermined coefficients).
Substituting this into the given differential equation gives
yh general solution of the homogeneous equation
Auxiliary Equation
m2 3m 10 0
m 5 m 2 0
m 5, 2
yh c1e5x c 2e2x
y Ax 2 Bx C
y ' 2Ax B
y " 2A
2A 3 2Ax B 10 Ax 2 Bx C 5x 2
1
x2 : 10A 5 A
2
3
x: 6A 10B 0 B
10
19
c: 2A 3B 10C 0 C
100
1 3 19
yp x 2 x
2 10 100
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DE 730 MW
y complete solution of nonhomogeneous linear differential equation
y yh yp
1 2 3 19
y c1e5x c 2e2x x x
2 10 100
Exercises
Find the complete solution of the following differential equation
1 7x
1. y " 6y ' 25y 8e 7x y e 3x c1 cos 4x c 2 sin 4x e
4
2. y " y ' 6y 10e3x y c1e3x c 2e 2x 2xe3x
1 1
3. y " y ' 6y e x 12x y c 1 e 3 x c 2 e 2 x e x –2 x
6 3
4. y' 8y 16; y 0 5 y 5e8x 2
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DE 730 MW
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