Q. 245.0( 2 Votes )

# If x = sin, show that (1–x^{2})y_{2}–xy_{1}–a^{2} y = 0

Answer :

Note: y_{2} represents second order derivative i.e. and y_{1} = dy/dx

Given,

x = sin

y = ……equation 1

to prove: (1–x^{2})y_{2}–xy_{1}–a^{2} y = 0

We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative.

Let’s find

As,

So, lets first find dy/dx

∵ y =

Let t = asin^{–1} x => []

And y = e^{t}

…….equation 2

Again differentiating with respect to x applying product rule:

Using chain rule and equation 2:

[using ]

Using equation 1 and equation 2 :

∴ (1–x^{2})y_{2}–xy_{1}–a^{2}y = 0……proved

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