The document discusses higher order non-homogeneous linear differential equations and methods for finding their particular integrals. It defines a general higher order non-homogeneous differential equation and explains that the general solution is the sum of the complementary solution and particular solution. It then presents three rules for finding the particular integral when the forcing term F(x) is an exponential, sine, or cosine function. The rules involve taking derivatives of the differential operator f(D) evaluated at constants related to the exponential, sine, or cosine function.

1. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
LECTURE #11
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 1:
1
𝑓(𝐷)
𝑒 𝑎𝑥
=
1
𝑓(𝑎)
𝑒 𝑎𝑥
If 𝑓(𝑎) = 0 then
1
𝑓(𝐷)
𝑒 𝑎𝑥
= 𝑥
1
𝑓′(𝑎)
𝑒 𝑎𝑥
If 𝑓′′(𝑎) = 0 then
1
𝑓(𝐷)
𝑒 𝑎𝑥
= 𝑥2
1
𝑓′′(𝑎)
𝑒 𝑎𝑥
And so on.
RULE 2:
1
𝑓(𝐷)
𝑠𝑖𝑛𝑎𝑥 =
1
𝑓(−𝑎2)
𝑠𝑖𝑛𝑎𝑥
If 𝑓(−𝑎2) = 0 then

2. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
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
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.