Partial derivatives: Problems under type-4

PROBLEMS UNDER TYPE 4

Let u= f(r,s,t). r, s and t are functions of x, y, z.Then

Example 34. If u = f(x‒y, yz, z‒x), then prove that ∂u/∂x + ∂u/∂y + ∂u/∂z =0.

Solution: Let r=x‒y, s=y‒z, tz‒x.

:. u = f(r,s,t)

Example 35. If u = f(2x‒3y, 3y‒4z, 4z‒2x), then prove that .

Solution: Let r = 2x‒3y, s = 3y‒4z, t = 4z ‒ 2x

u = f(r,s,t)

Example 36. If u = f(x/y, y/z, z/x), then prove that x.∂u/∂x + y.∂u/∂y + z.∂u/∂z = 0.

Solution:

Let r =x/y, s =y/z, t =z/x.

:. u = f(r,s,t)

Example 37. If u = u( y‒x / xy, z‒x / xz), then prove that .

Solution: 

Example 38. If u = f(x, y), where x = r cos 0, y = r sin 0, then P.T 

Solution.

Given x = r cos 0, y = r sin 0

Example 39. If Z = f (u,v), where u = lx + my, v = ly‒mx then prove that .

Solution: Given u = lx + my, v = ly ‒ mx

Example 40. Prove that , u = e^xcosy, v = e^xsiny and that f is a function of u and v and also of x and y.

Solution: Given u = e^xcosy, v = e^xsin y

Hence the proof

Example 41. If f = f(u, v), where u = x²‒y² and v = 2xy, then show that .

Solution: Given u = x² ‒ y²; v = 2xy

Example 42. Transform the equation ∂²z/∂x² + ∂²z/∂y² = 0 into polar co ordinates.

Solution: Polar co‒ordinate is x = r cos 0, y = r sine

x² + y² = r²

r = √[x² + y²]

y/x = tanθ

θ= tan^‒1x/y

EXERCISE

46. Find du/dx when u = sin(x² + y²), where x² + 4y² = 9.

Ans: 3xcos(x²+y²) / 2

47. Find du/dx when u = tan^‒1(y/x), where x² + y² = a².

Ans: ‒1/y

48. Find du/dt if u=x/y where x = e^t, y = log t.

Ans: dz/dt = e^t/logt [ 1‒ 1/tlogt ]

49. If z = x² + y², where x = t³, y = 1 + t², find dz/dt.

Ans: = dz/dt  = 6t⁶ + 4t + 4t³

50. If ax² + 2hxy + by² = 1, then prove that d²y/dx² = h²‒ab / (hx+by)³

51. Using partial Derivative find dy/dx for (sec x)^y = (cot y)^x.

Ans. [ log coty ‒ y tan x ] /  [log secx + x secy cosec y]

52. Using partial Derivative find dy/dx for (cos x)^y = (sin x)^y.

Ans. [ y tan x + log sin y ] / [ log cos x ‒ x coty ]

53. If ƒ (cx ‒ az, cy ‒ bz) = 0, where z is a function of x and y, prove that a.∂z/∂x + b. ∂z/∂y = c.

54. If ƒ is a function of u, v, w where u = √(yz), v = √(zx), w= √(xy) show that  

55. If u = u(x, y) and x = e^r cos θ, y = e^r sin θ, show that 

56. If z = f(u, v), where u = coshx cosy and v = sin hx sin y, then prove that 

57. Transform the equation , by changing the independent variables using u = 2x + y and v = 3x + y.

Ans: ∂²z/∂u∂u = 0

58. Using partial differentiation, prove d²y/dx² = b²‒ac / (ay+b)³ when ay²+2by + c = x².

59. If u = f(r,s), r=x+at, s= y + bt, then show that ∂u/∂t = a [∂u/∂x] + b [∂u/∂x]

60. If z = f(x, y) where, x = Xcosa ‒ Ysina and y = Xsina + Ycosa show .

61. Transform the equation  = 0 by changing the independent variables using u = x ‒ y and v = x + y.