If #y=x + sqrt(a^2 + x^2)# where #a# is a constant, prove that #(a^2 + x^2)(d^2y)/(dx^2) + xdy/dx - y = 0#?

1 Answer
Steve M
Apr 24, 2018

We have:

# y=x + sqrt(a^2 + x^2) #

Which we can write as:

# y - x = sqrt(a^2 + x^2) #

# :. (y - x)^2 = a^2 + x^2 #

# :. y^2-2xy + x^2 = a^2 + x^2 #

# :. y^2-2xy = a^2 #

Now, differentiating Implicitly we have:

# 2ydy/dx-2{xdy/dx+y} = 0 #

# :. ydy/dx - xdy/dx-y = 0 #

# :. (y-x)dy/dx = y => y(dy/dx-1)=xdy/dx#

Differentiating Implicitly again:

# (y-x)(d^2y)/(dx^2) + (dy/dx)(dy/dx-1) =dy/dx #

# (y-x)(d^2y)/(dx^2) + (dy/dx)(dy/dx-1) = y/(y-x) #

# :. (y-x)^2(d^2y)/(dx^2) + (y-x)(dy/dx)(dy/dx-1) = y #

# :. (a^2 + x^2)(d^2y)/(dx^2) + y(dy/dx-1) = y #

# :. (a^2 + x^2)(d^2y)/(dx^2) + xdy/dx = y #

# :. (a^2 + x^2)(d^2y)/(dx^2) + xdy/dx - y =0 \ \ \ # QED

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