Question

If \(x = sin(t),y = cos(pt)\), then \((1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx} =\)

• A. \(0\)
• B. \(-p^2y\)
• C. \(p^2y\)
• D. \(p\)

Hint:

Express dy/dx and d2y/dx2 using t, then substitute into the given expression.

Answer 1

The Correct Option is B

Step 1: Use y as a function of t:
Let $y=\cos(pt)$ so $\dfrac{d^2y}{dt^2}=-p^2y$.

Step 2: Convert d2y/dt2 into x-derivatives:
With $x=\sin t$: $\dfrac{dy}{dt}=\cos t\,y'$ and $\dfrac{d^2y}{dt^2}=\cos^2t\,y''-\sin t\,y'=(1-x^2)y''-xy'$.

Step 3: Equate:
$(1-x^2)y''-xy'=-p^2y$. Option B.

Final Answer:
Since d2y/dt2 = -p^2 y, the expression equals -p^2 y. \[ \boxed{\text{(B) }-p^2y} \]