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Chapter 6: Problem 114

If \(x=a \cos \theta, y=b \sin \theta\), find \(\frac{d^{2} y}{d x^{2}}\)

Short Answer

Expert verified
The second derivative of y with respect to x, \(d^2y/dx^2\), is \(\frac{b^2 - y^2}{b^2y})\).

Step by step solution

01

Calculate dy/dx

We first find the derivatives of x and y with respect to \(\theta\). Here, \(dx/d\theta= -a \sin\theta\) and \(dy/d\theta = b \cos\theta\). Now we can calculate \(dy/dx\) by using the chain rule \(dy/dx= (dy/d\theta)/(dx/d\theta)\) which gives us \(dy/dx = -(b \cos\theta)/(a \sin\theta)\)
02

Further simplify dy/dx

Now, we need to write \(dy/dx\) in terms of x. Observe that \(\sin\theta = y/b\) from the original equation given. Replace \(\sin\theta\) in the expression of \(dy/dx\) to obtain \(dy/dx = - (b \cos\theta)/a(y/b)\) which simplifies to \(dy/dx = -\cos\theta/a(y/b)\). Also, from the initial equations given, \(\cos\theta = x/a\). Substitute that into the equation to obtain \(dy/dx = -(x/a)/(a(y/b))\), which simplifies to \(dy/dx = -x/y\)
03

Compute d^2y/dx^2

The second derivative \(d^2y/dx^2\) is the derivative of \(dy/dx\) with respect to x. So, we will compute \(d^2y/dx^2\) by differentiating -x/y with respect to x. Note that -x/y can be rewritten as -x*y^-1. The derivative of a product of two functions is obtained by applying the product rule, which is \(d(uv)/dx = v(du/dx) + u(dv/dx)\). Therefore, \(d(-x*y^-1)/dx = -y^-1(dx/dx) - x (-y^-2)(dy/dx) = -y^-1 - x (-y^-2)(-x/y) = -y^-1 - x^2/y^3\)
04

Simplify d^2y/dx^2

Now let's simplify \(d^2y/dx^2\). Both terms in the expression share a common factor y^-1, thus the expression can be simplified by factoring out y^-1. So, \(d^2y/dx^2 = y^-1(1 - x^2/y^2) = y^-1(1 - x^2/b^2*sin^2\theta)\), which, utilizing sin^2胃 + cos^2胃 = 1, results in \(d^2y/dx^2= y^-1(1 - x^2/y^2) = y^-1(1 - y^2/b^2)\). Simplifying further to put it in standard form we get \(d^2y/dx^2 = \frac{b^2 - y^2}{b^2y}\)

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Most popular questions from this chapter

Find \(\frac{d^{2} y}{d x^{2}}\), if (i) \(x=a t^{2}, y=2 a t\) (ii) \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\) (iii) \(x=a \cos \theta, y=b \sin \theta\)

If \(y=c_{1} e^{x}+c_{2} e^{-x}\), then prove that \(\frac{d^{2} y}{d x^{2}}-y=0\).

If \(x \sqrt{1+y}+y \sqrt{1+x}=0\), prove that \(\frac{d y}{d x}=-\frac{1}{(1+x)^{2}}\)

If \(y=x \sin y\), prove that \(\frac{d y}{d x}=\frac{y}{x(1-x \cos y)} .\)

If \(a, b, c \in\left(0, \frac{\pi}{2}\right)\) and \(a
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