DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
LECTURE #10
HIGHER ORDER DIFFERENTIAL EQUATIONS:
HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS:
Consider the equations
π0
πππ¦
ππ₯π+ π1
ππβ1π¦
ππ₯πβ1+ β― + ππβ1
ππ¦
ππ₯+ πππ¦ = 0 (1)
Where π0, π1, β¦ , ππβ1,ππ are real constants.
To find the solution consider exponential function. Her we have for π¦ = πππ₯
ππ¦
ππ₯= ππππ₯,
π2π¦
ππ₯2= π2πππ₯, . . . ,
ππβ1π¦
ππ₯πβ1= ππβ1πππ₯,
πππ¦
ππ₯π= πππππ₯
Substituting in (1)
π0πππππ₯ , π1ππβ1πππ₯ + β― + ππβ1ππππ₯ + πππππ₯ = 0
πππ₯(π0ππ, π1ππβ1 + β― + ππβ1π + ππ) = 0
Since πππ₯ β 0
π0ππ, π1ππβ1 + β― + ππβ1π + ππ = 0 (2)
Thus π is a solution of (1)if and only if π is the solution (2).
Eqβ (2) is called the characteristics (or auxiliary) equation of the given differential
equation (1)
Three cases arise as the roots (2) are
(i) Real and distinct
(ii) Real and repeated
(iii) Complex
CASE I: DISTINCT REAL ROOTS
Let π1, π2, π3, β¦ , ππ be the β²πβ² real distinct roots then
π¦ = π1ππ1π₯ + π2ππ2π₯ + β― + ππβ1πππβ1π₯ + ππππππ₯
DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
CASE# II: REPEATED REAL ROOTS
Let π1, π2, π3, β¦ , ππ be the β²πβ² real distinct roots where π1 = π2then
π¦ = (π1 + π2)ππ1π₯ + β― + ππβ1πππβ1π₯ + ππππππ₯
CASE III: COMPLEX ROOTS:
Let π be a complex roots then
π¦ = πππ₯(π1π ππππ₯ + π2πππ ππ₯)
EXAMPLE #1: (π·2 + 4π· + 3)π¦ = 0
EXAMPLE #2: (π·3 β 5π·2 + 7π· β 3)π¦ = 0
EXAMPLE #3: (π·3 β π·2 + π· β 1)π¦ = 0
EXAMPLE #4: (π·2 + π· β 12)π¦ = 0
EXAMPLE #5: (π·2 + 4π· + 5)π¦ = 0
EXAMPLE #6: (π·3 β 3π·2 + 4)π¦ = 0
EXAMPLE#7: (9π·2 β 12π· + 4)π¦ = 0
EXAMPLE #8: (75π·2 + 50π· + 12)π¦ = 0
EXAMPLE #9: (π·3 β 4π·2 + π· + 6)π¦ = 0
EXAMPLE #10: (π·3 β 6π·2 + 12π· β 8)π¦ = 0
EXAMPLE #11: (π·3 β 27)π¦ = 0
EXAMPLE #12: (4π·4 β 4π·3 β 3π·2 + 4π· β 1)π¦ = 0
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