Various methods of finding the particular integrals:
1. When X = e^{ax} in f(D) y = X, where a is a constant
Then 1/f(D) e^{ax }= 1/f(a) e^{ax} , if f(a) ≠ 0 and
1/f(D) e^{ax} = x^{r}/f^{r}(a) e^{ax} , if f(a) = 0, where f(D) = (Da)^{r}f(D)
2. To find P.I. when X = cos ax or sin ax
f (D) y = X
If f (– a^{2}) ≠ 0 then 1/f(D^{2}) sin ax = 1/f(a^{2}) sin ax
If f (– a^{2}) = 0 then (D^{2} + a^{2}) is at least one factor of f (D^{2})
3. To find the P.I.when X = xm where m ∈ N
f (D) y = xm
y = 1/ f(D) xm
4. To find the value of 1/f(D) e^{ax} V where ‘a’ is a constant and V is a function of x
1/f (D) .e^{ax} V = e^{ax}.1/f (D+a). V
5. To find 1/f (D). xV where V is a function of x
1/f (D).xV = [x 1/f(D). f'(D)] 1/f(D) V
Some Results on Tangents and Normals:
1. The equation of the tangent at P(x, y) to the curve y= f(x) is Y – y = dy/dx .(Xx)
2. The equation of the normal at point P(x, y) to the curve y = f(x) isY – y = [1/ (dy/dx) ].(X – x )
3. The length of the tangent = CP =y √[1+(dx/dy)^{2}]
4. The length of the normal = PD = y √[1+(dy/dx)^{2}]
5. The length of the Cartesian sub tangent = CA = y dy/dx
6. The length of the Cartesian subnormal = AD = y dy/dx
7. The initial ordinate of the tangent = OB = y – x.dy/dx
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