Various methods of finding the particular integrals:
1. When X = eax in f(D) y = X, where a is a constant
Then 1/f(D) eax = 1/f(a) eax , if f(a) ≠ 0 and
1/f(D) eax = xr/fr(a) eax , if f(a) = 0, where f(D) = (D-a)rf(D)
2. To find P.I. when X = cos ax or sin ax
f (D) y = X
If f (– a2) ≠ 0 then 1/f(D2) sin ax = 1/f(-a2) sin ax
If f (– a2) = 0 then (D2 + a2) is at least one factor of f (D2)
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) eax V where ‘a’ is a constant and V is a function of x
1/f (D) .eax V = eax.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 .(X-x)
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
![]() |
Use Code STAYHOME200 and get INR 200 additional OFF
|
Use Coupon Code |
2 videos|258 docs|160 tests
|