Question

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`?

Answer 1

Please see below.

Explanation:

Here,

`y=x+sqrt(x^2+a^2)...tocolor(red)((A))`

Diff. w.r.t. `x`,we get

`(dy)/(dx)=1+1/(2sqrt(x^2+a^2))xx2x`

`=>(dy)/(dx)=1+x/sqrt(x^2+a^2)`

`"Multiplying both sides by "` `color(orange) sqrt(x^2+a^2)`

`=>sqrt(x^2+a^2)(dy)/(dx)=sqrt(x^2+a^2)+x...tocolor(red)([Use (A)]`

`=>sqrt(x^2+a^2)(dy)/(dx)=y ...tocolor(violet)((B)`

Again diff. w. r. t. `x`, `"using "color(blue)" Product Rule"`

#sqrt(x^2+a^2) (d^2y)/(dx^2)+ (dy)/(dx)*1/(2sqrt(x^2+a^2))xx2x=(dy)/(dx)#

`"Multiplying both sides by "` `color(orange) sqrt(x^2+a^2)`

#(x^2+a^2)(d^2y)/(dx^2)+(dy)/(dx)xxx=sqrt(x^2+a^2) (dy)/(dx)...tocolor(violet)([Use(B)]#

`(x^2+a^2)(d^2y)/(dx^2)+x(dy)/(dx)=y`

`(x^2+a^2)(d^2y)/(dx^2)+x(dy)/(dx)-y=0`