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

𝑑𝑛𝑦

𝑑𝑥𝑛+ 𝑎1

𝑑𝑛−1𝑦

𝑑𝑥𝑛−1+ ⋯ + 𝑎𝑛−1

𝑑𝑦

𝑑𝑥+ 𝑎𝑛𝑦 = 𝐹(𝑥)

Where 𝑎0, 𝑎1, … , 𝑎𝑛−1,𝑎𝑛 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

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.