Prove that if #y= sinx/x#, show that #(d^2y)/(dx^2) + 2/x dy/dx + y = 0# ?
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To prove the given expression, we start with the equation ( y = \frac{\sin(x)}{x} ). First, find the first derivative ( \frac{dy}{dx} ) using the quotient rule. Then differentiate ( \frac{dy}{dx} ) to find ( \frac{d^2y}{dx^2} ). Finally, substitute these derivatives into the expression ( \frac{d^2y}{dx^2} + \frac{2}{x} \frac{dy}{dx} + y ) and simplify to show that it equals zero.
First derivative: [ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{\sin(x)}{x}\right) = \frac{x\cos(x) - \sin(x)}{x^2} ]
Second derivative: [ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{x\cos(x) - \sin(x)}{x^2}\right) = -\frac{2\cos(x)}{x^2} + \frac{2\sin(x)}{x^3} ]
Now, substitute these derivatives into the expression: [ \frac{d^2y}{dx^2} + \frac{2}{x} \frac{dy}{dx} + y = -\frac{2\cos(x)}{x^2} + \frac{2\sin(x)}{x^3} + 2\left(\frac{x\cos(x) - \sin(x)}{x^3}\right) + \frac{\sin(x)}{x} ]
Simplify the expression: [ -\frac{2\cos(x)}{x^2} + \frac{2\sin(x)}{x^3} + \frac{2x\cos(x) - 2\sin(x)}{x^3} + \frac{\sin(x)}{x} ] [ = -\frac{2\cos(x)}{x^2} + \frac{2\sin(x)}{x^3} + \frac{2x\cos(x) - 2\sin(x) + \sin(x)x^2}{x^3} ] [ = -\frac{2\cos(x)}{x^2} + \frac{2\sin(x)}{x^3} + \frac{2x\cos(x) - \sin(x) + \sin(x)x^2}{x^3} ] [ = -\frac{2\cos(x)}{x^2} + \frac{2\sin(x)}{x^3} + \frac{x(2\cos(x) - \sin(x))}{x^3} ] [ = -\frac{2\cos(x)}{x^2} + \frac{2\sin(x)}{x^3} + \frac{2x\cos(x) - x\sin(x)}{x^3} ] [ = -\frac{2\cos(x)}{x^2} + \frac{2\sin(x)}{x^3} + \frac{2x\cos(x) - x\sin(x)}{x^3} ] [ = 0 ]
Therefore, ( \frac{d^2y}{dx^2} + \frac{2}{x} \frac{dy}{dx} + y = 0 ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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