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.

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