DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM

DIFFERENTIAL EQUATION (MT-202)

LECTURE #10

HIGHER ORDER DIFFERENTIAL EQUATIONS:

HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS:

Consider the equations

𝑎0

𝑑𝑛𝑦

𝑑𝑥𝑛+ 𝑎1

𝑑𝑛−1𝑦

𝑑𝑥𝑛−1+ ⋯ + 𝑎𝑛−1

𝑑𝑦

𝑑𝑥+ 𝑎𝑛𝑦 = 0 (1)

Where 𝑎0, 𝑎1, … , 𝑎𝑛−1,𝑎𝑛 are real constants.

To find the solution consider exponential function. Her we have for 𝑦 = 𝑒𝑚𝑥

𝑑𝑦

𝑑𝑥= 𝑚𝑒𝑚𝑥,

𝑑2𝑦

𝑑𝑥2= 𝑚2𝑒𝑚𝑥, . . . ,

𝑑𝑛−1𝑦

𝑑𝑥𝑛−1= 𝑚𝑛−1𝑒𝑚𝑥,

𝑑𝑛𝑦

𝑑𝑥𝑛= 𝑚𝑛𝑒𝑚𝑥

Substituting in (1)

𝑎0𝑚𝑛𝑒𝑚𝑥 , 𝑎1𝑚𝑛−1𝑒𝑚𝑥 + ⋯ + 𝑎𝑛−1𝑚𝑒𝑚𝑥 + 𝑎𝑛𝑒𝑚𝑥 = 0

𝑒𝑚𝑥(𝑎0𝑚𝑛, 𝑎1𝑚𝑛−1 + ⋯ + 𝑎𝑛−1𝑚 + 𝑎𝑛) = 0

Since 𝑒𝑚𝑥 ≠ 0

𝑎0𝑚𝑛, 𝑎1𝑚𝑛−1 + ⋯ + 𝑎𝑛−1𝑚 + 𝑎𝑛 = 0 (2)

Thus 𝑚 is a solution of (1)if and only if 𝑚 is the solution (2).

Eq→ (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 𝑚1, 𝑚2, 𝑚3, … , 𝑚𝑛 be the ′𝑛′ real distinct roots then

𝑦 = 𝑐1𝑒𝑚1𝑥 + 𝑐2𝑒𝑚2𝑥 + ⋯ + 𝑐𝑛−1𝑒𝑚𝑛−1𝑥 + 𝑐𝑛𝑒𝑚𝑛𝑥

DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM

DIFFERENTIAL EQUATION (MT-202)

CASE# II: REPEATED REAL ROOTS

Let 𝑚1, 𝑚2, 𝑚3, … , 𝑚𝑛 be the ′𝑛′ real distinct roots where 𝑚1 = 𝑚2then

𝑦 = (𝑐1 + 𝑐2)𝑒𝑚1𝑥 + ⋯ + 𝑐𝑛−1𝑒𝑚𝑛−1𝑥 + 𝑐𝑛𝑒𝑚𝑛𝑥

CASE III: COMPLEX ROOTS:

Let 𝑚 be a complex roots then

𝑦 = 𝑒𝑎𝑥(𝑐1𝑠𝑖𝑛𝑏𝑥 + 𝑐2𝑐𝑜𝑠𝑏𝑥)

EXAMPLE #1: (𝐷2 + 4𝐷 + 3)𝑦 = 0

EXAMPLE #2: (𝐷3 − 5𝐷2 + 7𝐷 − 3)𝑦 = 0

EXAMPLE #3: (𝐷3 − 𝐷2 + 𝐷 − 1)𝑦 = 0

EXAMPLE #4: (𝐷2 + 𝐷 − 12)𝑦 = 0

EXAMPLE #5: (𝐷2 + 4𝐷 + 5)𝑦 = 0

EXAMPLE #6: (𝐷3 − 3𝐷2 + 4)𝑦 = 0

EXAMPLE#7: (9𝐷2 − 12𝐷 + 4)𝑦 = 0

EXAMPLE #8: (75𝐷2 + 50𝐷 + 12)𝑦 = 0

EXAMPLE #9: (𝐷3 − 4𝐷2 + 𝐷 + 6)𝑦 = 0

EXAMPLE #10: (𝐷3 − 6𝐷2 + 12𝐷 − 8)𝑦 = 0

EXAMPLE #11: (𝐷3 − 27)𝑦 = 0

EXAMPLE #12: (4𝐷4 − 4𝐷3 − 3𝐷2 + 4𝐷 − 1)𝑦 = 0