The document discusses higher order non-homogeneous linear differential equations, outlining their general solution as the sum of the complementary and particular solutions. It presents rules for finding the particular integral, including relationships involving sine and cosine functions. The equations are illustrated through specific mathematical expressions and conditions.

1. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
LECTURE #12
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 2:
1
𝑓(𝐷)
𝑠𝑖𝑛𝑎𝑥 =
1
𝑓(−𝑎2)
𝑠𝑖𝑛𝑎𝑥
If 𝑓(−𝑎2) = 0 then
1
𝑓(𝐷)
𝑠𝑖𝑛𝑎𝑥 = 𝑥
1
𝑓′(−𝑎2)
𝑠𝑖𝑛𝑎𝑥
If 𝑓′(−𝑎2) = 0 then
1
𝑓(𝐷)
𝑠𝑖𝑛𝑎𝑥 = 𝑥2
1
𝑓′′(−𝑎2)
𝑠𝑖𝑛𝑎𝑥
And so on.
Similarly,
1
𝑓(𝐷)
𝑐𝑜𝑠𝑎𝑥 =
1
𝑓(−𝑎2)
𝑐𝑜𝑠𝑎𝑥
If 𝑓(−𝑎2) = 0 then

2. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
1
𝑓(𝐷)
𝑐𝑜𝑠𝑎𝑥 = 𝑥
1
𝑓′(−𝑎2)
𝑐𝑜𝑠𝑎𝑥
If 𝑓′(−𝑎2) = 0 then
1
𝑓(𝐷)
𝑐𝑜𝑠𝑎𝑥 = 𝑥2
1
𝑓′′(−𝑎2)
𝑐𝑜𝑠𝑎𝑥
And so on.