Special Form of X in Differential Equation
Special Form of X in Differential Equation:
Type 2: P.I of the form
We have D ( Sin ax) = a cos ax
Therefore, if f (D2) is a rational integer function of D2 then f(D2) sin ax = f(-a2) sin ax.
f ( - a2) ≠ 0
Similarly we can provide that,
The formula can be easily remembered as follows:
Type 3: P.I of the form Φx/ f(D) where Φ(x) is a polynomial in x, we seeking the polynomial equation as the particular solution of :
f (D)y = Φ (x)
Hence P.I os found by divisor. By writing Φ (x) in decending power of x and f(D) in ascending power of D. The division get completed without any remainder. The quotient so obtained in the process of division will be particular integral.