Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th

# If y = (x+1+x2)m, then show that (1 + x2) y2 + xy1 – m2y = 0 - Business Mathematics and Statistics

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Sum

If y = (x + sqrt(1 + x^2))^m, then show that (1 + x2) y2 + xy1 – m2y = 0

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#### Solution

y = (x + sqrt(1 + x^2))^"m"

y_1 = "m"(x + sqrt(1 + x^2))^("m"-1) {1 + (2x)/(2sqrt(1 + x^2))}

= "m" (x + sqrt(1 + x^2))^("m" - 1){(sqrt (1+ x^2) + x)/(sqrt(1 + x^2))} = ("m"(x + sqrt(1 + x^2))^"m")/(sqrt(1 + x^2))

y_1 = "my"/sqrt(1 + x^2)

Squaring both sides we get,

y_1^2 = ("m"^2y"^2)/(1 + x^2)

(1 + x2) (y_1^2) = m2y2

Differentiating with respect to x, we get

(1 + x2) . 2(y1) (y2) + (y1)2 (2x) = 2m2yy1

Dividing both sides by 2y1 we get,

(1 + x2) y2 + xy1 = m2y

⇒ (1 + x2) y2 + xy1 – m2y = 0

Concept: Differentiation Techniques
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