Tagged Questions

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Differentiate the function

$$v=\left(\sqrt{x} + {1\over x^{1\over 3}} \right)^2$$ We are working on differentiating functions. This one I have tried everything on and my teacher keeps saying I'm wrong. I'm just not seeing what ...
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Inverse of $r sin(\omega t) + v t$?

I am wondering if there is an inverse for this function, $x(t)=r sin(\omega t) + v t$. The inverse function theorem suggests that an inverse for this function does exist, although it may have to be ...
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Simplifying a coupled-pendulum equation by assumption

I have been given the following question and I am unsure if I am missing an assumption or if I am misunderstanding something else: Two identitical pendula each of length $\ell$ and with bobs of ...
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How can I derive differential equations for polar coordinates based on these equations?

A textbook I am using on my own to study differential equations contains a problem: given the two differential equations for $x,y$ below and a real value of $t$, derive the differential equations for ...
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How do I prove that both solutions to this differential equation $y"+k^2y=0$ are equivalent?

Consider the following differential equation $y''+k^2y=0$, where $y''$ is the 2nd derivative of y with respect to x. The solution to this equation is $y = A\exp(ikx) + B\exp(-ikx)$. However, another ...
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Differential equation trouble

I am trying to solve the following differential equation: $$\frac{\mathrm{d}y}{\mathrm{d}x}=2(2x+y)^2$$ If we make the substitution $z=2x+y$, then we get: ...
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Differentiate the functions trigonometry and derivatives

Differentiate $ln(\sec x+ \tan x)$ and $(\sin x)^3\cos 3x+ (\cos x)^3\sin 3x$ with respect to $x$, simplifying where possible. Find the first and second derivatives (with respect to x) of the ...
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How to treat system of linear first order differential equations with trigonometric function coefficients?

I'm having trouble solving the following IVP: $$x_1^\prime = -x_1\tan t + 3\cos^2t$$ $$x_2^\prime = x_1 + x_2\tan t + 2\sin t$$ where $x_1(0) = 4$ and $x_2(0) = 0$. I'm not sure what to do when the ...
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Second Order Linear Differential Equations with Constant Coefficients Containing Trigonometric Functions

I'm having trouble applying the method of undetermined coefficients, as explained in Apostol's Calculus, to second order linear differential equations with constant coefficients containing ...
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Laplace transform of trig + Heaviside

So I am trying to take the laplace transform of $\cos(t)u(t-\pi)$. Is it valid for me to treat it as $((\cos(t)+\pi)-\pi)u(t-\pi)$ and treat $\cos(t)-\pi$ as $f(t)$ and use the 2nd shifting property, ...
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Why does using different units of angle affect the rate of change?

The question is A triangle has sides of length 4cm and 9cm. The angle between them is increasing at a rate of 1$^\circ$ per minute. Find the rate in cm$^2$ per minute at which the area of the ...
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Integrate $\int\frac{Cx}{(\sin x^2)^2}dx$

Have been a doing a reduction of order ODE problem and this integral comes up at the last step. Not sure how to go about integrating it. The answers give $\cos x^2$ as the answer. Here's the original ...
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A Functional Differential Equation

I was having a play with some trig. identities and noticed the following: $\cos{x}=\frac{\sin{2x}}{2\sin{x}}$ Now, $\cos{x} = \frac{d}{dx}\sin{x}$ so I made the following analogous differential ...
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Second order linear ODE with trigonometric coefficient

Is there a theory and a name for the second order linear ODE with trigonometric coefficient (other than the Floquet theory)? The equation in question, with $a$,$b$,$c$ periodic function containing ...
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Calculate the $\int_0^{2\pi}\cos(mx)\cos(nx)dx$

I'm having trouble with this problem: Consider the integral: $$\tag 1\int_0^{2\pi}\cos(mx)\cos(nx)dx$$ a. Write $\cos(mx)$ and $\cos(nx)$ in terms of complex exponentials and compute ...
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On the differential equation $y''+y=0$

Consider the differential equation $$\frac{d^{2}y}{dx^{2}}+y=0$$ with initial conditions $y(0)=0$ and $y'(0)=1$. The solution is well known - $y=\sin(x)$. I know how to derive this solution, since the ...
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Can you help me solve this ODE?

I need to solve this differential equation. YWhat I'm looking for is a way to simplify this equation. Can anybody give me hints/tricks to understand the following equation better: ...
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simple question involving trigonometry

Can anybody explain what $$\tan(\sin^{-1}(\frac x y))$$ equals? I have to determine whether $$y'' \left(\tan \left(\sin^{-1}\left(\frac x y \right)\right) - \frac{x}{\sqrt{y^2-x^2}} \right)=0$$ is a ...
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Trigonometric general solution to ordinary differential equation

Solve: $$\frac{dx}{dy}=(x^{2}-x-12)(1+\tan^{2}{y})$$ This is a first order, linear, separable ODE, so it can be solved by rearranging to: $$\frac{dx}{x^{2}-x-12}=(1+\tan^{2}{y})\:dy$$ And then ...
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Tenenbaum and Pollard, Ordinary Differential Equations, problem 1.4.29, what am I missing?

Tenenbaum and Pollard's "Ordinary Differential Equations," chapter 1, section 4, problem 29 asks for a differential equation whose solution is "a family of straight lines that are tangent to the ...
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Solve $\ddot\theta +k\sin(2\theta)=0$ given initial value and constraints

How is it possible to deduce from the equation $$\ddot\theta +k\sin(2\theta)=0$$ where $\theta=\theta(t)$ and $\tan(\theta)={b(t)\over a(t)}$, $k$ is constant, and $a(0)=a_0$, $a(t)^2+ b(t)^2=a_0^2$. ...
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A simple question about sine and cosine

I have been thinking about all of the different ways that I have encountered sine and cosine in my studies. There are no courses on trigonometry at my school, so perhaps that's why I feel like ...
$\frac{1}{3}\cos^3 x \cos(2x)+\frac{1}{12}\sin(2x)(\sin(3x)+3\sin x)=\frac{1}{3} \cos x$ I got this as the result of a differential equation that I solved. The answer in the book is (1/3) cos(x), but ...
I'm reading a text on ray tracing. There is this section about radiometric quantities where radiance is defined as $L = \frac{d^2\Phi}{dA cos\Theta d\omega}$ $\Phi$ is the radiant flux $\Theta$ is ...