1
vote
1answer
25 views

Diff. Eq. Example with Matrices

I'm currently working on a side project of mine that deals with $\sin(A)$ and $\cos(B)$, where $A,B\in\mathbb{C}^{nxn}$. I'm trying to find some interesting (or non-interesting) examples where one ...
1
vote
0answers
24 views

Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
2
votes
1answer
31 views

Dirac Delta Function, Initial Value Problem

Hi I finished this IVP but I cant seem to get the right answer can someone give me some advice as to where I went wrong and point me in the right direction as to how to fix it. Here is the problem and ...
2
votes
1answer
41 views

How to solve this differential equation sinusoidal?

I can't find how to separate variables. $$y= \sin(xy')$$
3
votes
3answers
101 views

Need help with simple system of differential equations

thanks to your help I advanced in computing differential equations, but now I encountered another problem I need help with - this time it is a system of differential equations: $$x_1'=-x_2$$ ...
0
votes
1answer
50 views

finding an explicit formula from arctangent

Seems straight-forward but i can't get it right; I have this implicit equation: $$-2 \arctan{ \left( \frac{\sqrt{y-y^2}}{y} \right) } -x=c$$ where c is a constant. I've to find the explicit ...
0
votes
0answers
30 views

Different ways to derive differential equations for sech/csch/sec/csc

I recently realised that I only know one way to derive the differential equations for the reciprocals trigs (the hyperbolic and normal secant and cosecant): brute search with taking derivatives. As ...
0
votes
1answer
15 views

Limit when $y>>a$ of a derived solution

I am able to do part d), however I am very stuck on part e). If $y >> a$ then surely we get $\phi(x,y)$ $= \frac{1}{\pi} \Big[ tan^{-1} \Big( \frac{x+a}{y}\Big)-tan^{-1} ...
0
votes
0answers
22 views

“Differential variations”?

This passage in an old book on trigonometry calls these relations among parts of a spherical triangle "differential variations". The "parts" are three sides and the three angles; when the sides are ...
2
votes
3answers
75 views

Ordinary differential equation $y'(t)=\sin(f(t,y))$

One whose solution never makes me happy is the following: $$y'(t)=\sin(y+t)\text{.}$$ I would start by substituting $z(t)=y(t)+t$ to get an ODE in $z(t)$, but then I'm not sure about how to substitute ...
0
votes
1answer
27 views

Solving a system of trig equations

My book somehow got from those top two equations in the picture and solved for v'2(t) + v'1(t). I don't see how they did this? Can anyone see the trick for this type of problem?
0
votes
1answer
42 views

solving a trigonometric differential equation

could anybody please show me how to solve $$y''+ 2y'+2y = 10 \cos(t)\ ?$$ I found the $Y_c$ , but I have problems figuring out the particular solution. Thank you for your help.
0
votes
2answers
56 views

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 ...
0
votes
2answers
57 views

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 ...
0
votes
1answer
52 views

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 ...
1
vote
1answer
88 views

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 ...
0
votes
1answer
80 views

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 ...
4
votes
4answers
81 views

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: ...
-2
votes
1answer
35 views

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 ...
0
votes
1answer
151 views

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 ...
0
votes
1answer
278 views

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 ...
1
vote
2answers
308 views

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, ...
0
votes
1answer
268 views

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 ...
0
votes
1answer
66 views

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 ...
3
votes
1answer
136 views

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 ...
1
vote
0answers
75 views

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 ...
12
votes
3answers
175 views

Can all points in the plane be represented like this?

Solving a task regaring affine geometry, I've come across a problem: Is it true that, for every point $(x,y)\in \mathbb{R}^2$, there exist $t\in \mathbb{R}, \alpha\in[0,2\pi]$, such that $$x = ...
0
votes
2answers
61 views

the period of a trigonometric function

I'm trying to solve a differential equation which is : $$y'(t)-4y(t) = \cos(3t)$$ Resolution of the equation without the second membre $y'(t)-4y(t)=0$ has as solution $ y_s(t)=ke^{4t} $ with ...
0
votes
1answer
50 views

Prove that $x^r$ with $r=a+ib$ or $r=a-ib$ defines real solutions

Prove that $x^r$ with $r=a+ib$ or $r=a-ib$ defines real solutions in terms of trigonometric functions with argument $\ln x$ multiplied by exponential function $y(x)=x^{(a+ib)x}$ or $y(x)=x^{(a-ib)x}$ ...
1
vote
2answers
29 views

Trigonometry in the DE

Just a simple undergrad physics student asking them mathematicians. I have a very simple 2nd order homogeneous DE of the form: $$y''=-a^2y$$ So, a solution will be of the form $$y(t)=c_1 ...
4
votes
3answers
737 views

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 ...
4
votes
1answer
131 views

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 ...
1
vote
2answers
63 views

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: ...
2
votes
2answers
69 views

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 ...
2
votes
2answers
357 views

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 ...
3
votes
1answer
330 views

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 ...
4
votes
1answer
230 views

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 $a(0)=a_0$, $a(t)^2+ b(t)^2=a_0^2$. ...
2
votes
1answer
93 views

Trignometric shifting in ODE. Wolframalpha gives different answer?

The ODE looks very identical ( kinda of) The ODE I have is $$y'' - y' + y = 0$$ $$y(0) = 5$$ $$y(1) = y(-1)$$ The solution (nontrivial) I got is $$y = 5e^{t/2}[\cos(\sqrt{3}x/2) + \left ( ...
2
votes
3answers
112 views

Some double angle identity to solve $(2x^{2}+y^{2})\frac{dy}{dx}=2xy$?

For some reason, I cannot see a clever way to solve this (I know the way of doing it like in Wolframalapha) but I am pretty sure there is a double angle identity to crack this puzzle. Could someone ...
7
votes
1answer
237 views

A higher-order differential equation involving absolute values and trigonometry

For a smooth function $f: (-\pi/2,\pi/2) \to \mathbb{R} $, if $\displaystyle\frac{|f''(x)|}{\sqrt{(1+f'(x))^3}} = \cos{x}$, and $f(0) = f'(0) = 0$, $f''(0) = 1$, ...
1
vote
1answer
194 views

A boundary value problem over an infinite interval

This is the edited version of the original problem, hopefully presented in a clearer manner. (I have also renamed this post with a more befitting title) Problem: $$y'(x) = ...
7
votes
3answers
481 views

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 ...
4
votes
2answers
204 views

Simplifying trigonometric expressions, is there a unified theory?

$\frac{1}{3}\cos^3 x \cos(2x)+\frac{1}{12}\sin(2x)(\sin(3x)+3\sin x)=\frac{1}{3} \cos x$ I got this as the result of a differential equation that I solved. The answer in the book is (1/3) cos(x), but ...
2
votes
1answer
746 views

What does the little d and d^2 mean in equations?

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 ...