Given \(\frac{d^2y}{dx^2}=cot\,x\frac{dy}{dx}+4y\,cosec^2x=0\). Changing the independent variable x to z by the substitution z=log tan \(\frac{x}{2}\), the equation is changed to
\(\frac{d^2y}{dz^2}+\frac{3}{y}=0\)
- \(2\frac{d^2y}{dz^2}+e^y=0\)
- \(\frac{d^2y}{dz^2}-4y=0\)
- \(\frac{d^2y}{dz^2}+4y=0\)
The Correct Option is C
Solution and Explanation
we need to simplify this equation in terms of z by expressing cot(x), sec^2(x/2), cosec^2(x), and their derivatives in terms of z using the given substitution z = log(tan(x/2)).
This simplification process requires rewriting these trigonometric functions in terms of tan(x/2) and then expressing tan(x/2) in terms of z.
At this point, the equation can be quite complex due to the trigonometric functions and their derivatives involved. The goal is to express everything in terms of z, simplify as much as possible, and obtain an equation involving y, z, and their derivatives.
The correct answer is option (C): \(\frac{d^2y}{dz^2}-4y=0\)
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Concepts Used:
Second-Order Derivative
The Second-Order Derivative is the derivative of the first-order derivative of the stated (given) function. For instance, acceleration is the second-order derivative of the distance covered with regard to time and tells us the rate of change of velocity.
As well as the first-order derivative tells us about the slope of the tangent line to the graph of the given function, the second-order derivative explains the shape of the graph and its concavity.
The second-order derivative is shown using \(f’’(x)\text{ or }\frac{d^2y}{dx^2}\).