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If y=log (x+ sqrt ( x^2+1)), prove that:...

If `y=log (x+ sqrt ( x^2+1)), prove that: (x^2+1) d^2y/dx^2 + xdy/dx=0`.

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PRADEEP PUBLICATION-CONTINUITY AND DIFFERENTIABILITY-EXERCISE
  1. Find the second derivative of log x + cosx.

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  2. If y = tan x + sec x, prove that (d^2y)/(dx^2) = (cosx)/((1-sinx)^2)

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  3. If y=log (x+ sqrt ( x^2+1)), prove that: (x^2+1) d^2y/dx^2 + xdy/dx=0.

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  4. Find the second order derivative of e^(x tanx)

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  5. If y = tan^-1 x , find (d^2y)/(dx^2) in terms of y alone.

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  6. If y = A sinx + B cosxthen prove that d^2y/dx^2 + y = 0

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  7. If y = 3 cos (log x) + 4 sin (log x) show that x^2 y2 + xy1 + y = 0

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  8. If y = Ae^(mx) + Be^(nx), Show that (d^2 /dx^2 y)-(m+n)dy/dx+mny = 0

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  9. If y=e^(ax)cosbx, then prove that (d^(2)y)/(dx^(2))-2a(dy)/(dx)+(a^(2)...

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  10. If y = (sqrt(x+1) - sqrt(x-1)), then prove that (x^2-1) y2 +xy1 = 1/4 ...

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  11. y = e^(m sin^(-1)x), prove that (1 - x^2) y2 - xy1 = m^2y

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  12. If x^2/a^2 - y^2/b^2 = 1, prove that ((d^2y)/dx^2) = (-b^4)/(a^2y^3)

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  13. If (x-a)^2 + (y-b)^2 = c^2 , for some c > 0, prove that [1+(dy/dx)^2]...

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  14. Find (d^2y)/dx^2, if x = a (theta - sin theta), y = a(1+cos theta)

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  15. If x = 2 cos theta - cos 2 theta, y = 2 sin theta - sin 2 theta, find ...

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  16. If x = a { cos t + log|tan t/2|} and y = a sin t, 0 < t < pi/2, find (...

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  17. If x=acos theta+bsintheta and y=a sintheta-bcostheta, then prove that ...

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  18. If y = x log (x/(a+bx)), prove that x^3 y 2 = (xy1 - y)^2

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  19. If x = g(t) and y = f(t), find (d^2y)/(dx^2) as a function of t.

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  20. If f(x) = |x|^3 show that f''(x) exists for all real x and find it.

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