differential equation Lecture#13
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The document discusses higher order non-homogenous linear differential equations, expressing the general solution as the sum of a complementary solution and a particular solution. It provides rules for finding particular integrals, including the application of binomial expansions. Additionally, it includes examples illustrating the solution of specific differential equations.
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differential equation Lecture#13
- 1. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM DIFFERENTIAL EQUATION (MT-202) LECTURE #13 HIGHER ORDER DIFFERENTIAL EQUATIONS: NON-HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS: The equations 𝑎0 𝑑 𝑛 𝑦 𝑑𝑥 𝑛 + 𝑎1 𝑑 𝑛−1 𝑦 𝑑𝑥 𝑛−1 + ⋯ + 𝑎 𝑛−1 𝑑𝑦 𝑑𝑥 + 𝑎 𝑛 𝑦 = 𝐹(𝑥) Where 𝑎0, 𝑎1, … , 𝑎 𝑛−1, 𝑎 𝑛 are real constants is called the higher order non- homogenous differential equation and its general solution is compose to two parts. i.e. General solution = Complimentary solution + Particular Solution RULES TO FIND PARTICULAR INTEGRAL: RULE 3: FOR PARTICULAR INTEGRAL OF POLYNOMIALS The following binomial expansions will be useful in this connection 1 1 − 𝑥 = 1 + 𝑥 + 𝑥2 + 𝑥3 + 𝑥4 + ⋯ 1 1 + 𝑥 = 1 − 𝑥 + 𝑥2 − 𝑥3 + 𝑥4 + ⋯ EXAMPLE #1: Solve (𝐷3 − 2𝐷 + 1)𝑦 = 2𝑥3 − 3𝑥2 + 4𝑥 + 5 EXAMPLE #2: Solve (𝐷4 + 8𝐷2 − 9)𝑦 = 9𝑥3 + 5𝑐𝑜𝑠2𝑥 EXAMPLE #3: Solve (𝐷4 − 2𝐷3 + 𝐷)𝑦 = 𝑥4 + 3𝑥 + 1