Question

What is the value of (x + y)^2y” if x = e^t sint and y = e^t cost?

(a) x/2(y’ + y)

(b) x/2(y’ – y)

(c) 2(xy’ + y)

(d) 2(xy’ – y)

I got this question in examination.

This key question is from Second Order Derivative in chapter Limits and Derivatives of Mathematics – Class 11

Answer 1

The correct answer is (d) 2(xy’ – y)

The best I can explain: Since, x = e^t sint and y = e^t cost

Therefore, dx/dt = e^t sint + e^t cost

= y + x

And, dy/dt = e^t cost – e^t sint

= y – x

So, y’ = dy/dx = (dy/dt)/(dx/dt) = (y – x)/(y + x)

Thus, y” = [(x + y)(y’ – 1) – (y – x)(y’ + 1)]/(x + y)^2

Or, (x + y)^2y” = (x + y – y + x)y’ – x – y + x

= 2xy’ – 2y

= 2(xy’ – y)