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differential equation Lecture#10
The document discusses higher order homogeneous linear differential equations, focusing on the general form and methods for finding solutions using exponential functions. It introduces the characteristic (or auxiliary) equation, detailing cases with real distinct, repeated, and complex roots. Various examples of differential equations are provided to illustrate these concepts.
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differential equation Lecture#10
- 1. 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 𝑑 𝑛 𝑦 𝑑𝑥 𝑛 + 𝑎1 𝑑 𝑛−1 𝑦 𝑑𝑥 𝑛−1 + ⋯ + 𝑎 𝑛−1 𝑑𝑦 𝑑𝑥 + 𝑎 𝑛 𝑦 = 0 (1) Where 𝑎0, 𝑎1, … , 𝑎 𝑛−1, 𝑎 𝑛 are real constants. To find the solution consider exponential function. Her we have for 𝑦 = 𝑒 𝑚𝑥 𝑑𝑦 𝑑𝑥 = 𝑚𝑒 𝑚𝑥 , 𝑑2 𝑦 𝑑𝑥2 = 𝑚2 𝑒 𝑚𝑥 , . . . , 𝑑 𝑛−1 𝑦 𝑑𝑥 𝑛−1 = 𝑚 𝑛−1 𝑒 𝑚𝑥 , 𝑑 𝑛 𝑦 𝑑𝑥 𝑛 = 𝑚 𝑛 𝑒 𝑚𝑥 Substituting in (1) 𝑎0 𝑚 𝑛 𝑒 𝑚𝑥 , 𝑎1 𝑚 𝑛−1 𝑒 𝑚𝑥 + ⋯ + 𝑎 𝑛−1 𝑚𝑒 𝑚𝑥 + 𝑎 𝑛 𝑒 𝑚𝑥 = 0 𝑒 𝑚𝑥(𝑎0 𝑚 𝑛 , 𝑎1 𝑚 𝑛−1 + ⋯ + 𝑎 𝑛−1 𝑚 + 𝑎 𝑛) = 0 Since 𝑒 𝑚𝑥 ≠ 0 𝑎0 𝑚 𝑛 , 𝑎1 𝑚 𝑛−1 + ⋯ + 𝑎 𝑛−1 𝑚 + 𝑎 𝑛 = 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, … , 𝑚 𝑛 be the ′𝑛′ real distinct roots then 𝑦 = 𝑐1 𝑒 𝑚1 𝑥 + 𝑐2 𝑒 𝑚2 𝑥 + ⋯ + 𝑐 𝑛−1 𝑒 𝑚 𝑛−1 𝑥 + 𝑐 𝑛 𝑒 𝑚 𝑛 𝑥
- 2. DIFFERENTIAL EQUATION (MT-202)SYED AZEEM INAM DIFFERENTIAL EQUATION (MT-202) CASE# II: REPEATED REAL ROOTS Let 𝑚1, 𝑚2, 𝑚3, … , 𝑚 𝑛 be the ′𝑛′ real distinct roots where 𝑚1 = 𝑚2then 𝑦 = (𝑐1 + 𝑐2)𝑒 𝑚1 𝑥 + ⋯ + 𝑐 𝑛−1 𝑒 𝑚 𝑛−1 𝑥 + 𝑐 𝑛 𝑒 𝑚 𝑛 𝑥 CASE III: COMPLEX ROOTS: Let 𝑚 be a complex roots then 𝑦 = 𝑒 𝑎𝑥(𝑐1 𝑠𝑖𝑛𝑏𝑥 + 𝑐2 𝑐𝑜𝑠𝑏𝑥) 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