Tagged Questions
1
vote
1answer
30 views
Simplifying Differentiation
I'm working off my textbook and I've followed the steps easily enough until it gets to this
$ \dfrac {dy}{dx} = \dfrac{(x^2 + 1)^3}{2 \sqrt{x - 1}} + \sqrt{x-1}~(6x)~(x^2 + 1)^2$
$= ...
4
votes
3answers
102 views
Where to start when learning math (again)?
I have a few questions I hope you can help me answer.
First, I'll introduce myself. I'm a finance undergraduate student in Australia, but I'm originally from Norway. Throughout school I always loved ...
0
votes
3answers
71 views
Does this IVP have a unique solution for all $x \in \mathbb R$
Is $\displaystyle {dy\over dx}=\sin(y)$ with initial conditions $y(X)=Y$ guaranteed to have a unique solution for all $x\in\mathbb R$?
2
votes
4answers
65 views
initial value problem: y'' + 4y = f(t) , y(0)= y'(0)=0. f(t) = { 0 if t <3; t if t >3}
Solve the initial value problem:
$$y'' + 4y = f(t) , y(0)= y'(0)=0. $$
where
$$ f(t) = \begin{cases} 0 &t < 3 \\ t & t > 3\end{cases} $$
I've solved for the homogeneous equation, $y'' ...
1
vote
0answers
21 views
Derivative methods for artifical neural networks with single hidden layer
I am trying to optimize the output of a given neural network with a single hidden layer. To accomplish this, I intend to find solve for all combinations of inputs where the derivative of the neural ...
5
votes
1answer
60 views
Having trouble using eigenvectors to solve differential equations
The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix}
5 & 4 \\
-1 & 1\\
\end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\
x_2 \\ \end{pmatrix}$$
I went ...
1
vote
0answers
115 views
“Two-speed” linear integro-differential equation
Working on a problem of many-electron dynamics in quantum dots I have arrived to an a following integro-differential equation:
$$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 ...
0
votes
1answer
44 views
Second Order Non-Linear ODE involving Bessel Functions
I'm trying to solve this but I'm getting nowhere. Does anyone know step-by-step solution? or at least the general techniques to use? I do know that the solution involves the Bessel functions.
$y'' + ...
0
votes
1answer
17 views
System of Differential Equations Question Assistance
The following question has just left me confused with no real decent avenue of attack so any assistance on this would be appreciated.
For the system of equations
$t {\frac{d \vec x}{dt}} = A\vec x $
...
1
vote
0answers
22 views
Reduction of order to find general solution of linear nonhomogenous differential equations. [closed]
Find the general solution of y''+p(t)y'+q(t)y=g(t) assuming y=v(t)y(t) is a solution. Use reduction of order to find the general solution.
0
votes
1answer
27 views
Curious about the differing limits of integration when solving an ODE
This is one of those things that I just haven't thought about but always used. It came up when I was looking over this old problem to solve $x′′+ϕ(x)=0$ with IC $x(0)=x0>0$ and $x′(0)=0$. But even ...
1
vote
2answers
72 views
resources to study PDE from
I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable ...
1
vote
1answer
58 views
How to solve a tensor differential equation?
Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$
where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor.
The original Problem
How does ...
0
votes
1answer
94 views
Solving $dP/dt = P(t)(1-P(t)/e^{2t})$ using $P(t) = 1/v(t)$ as a substitute
Basically, I tried solving $dP/dt = P(t)(1-P(t)/e^{2t})$ using $P(t) = 1/v(t)$ as a substitute, and got $P(t)=-e^{2t}+ce^t$ (where $c$ is a real constant). I got here by deriving the substitution and ...
1
vote
2answers
63 views
Solving $y'=\frac{1 + y^2 - x^2}{2 x y}$
I need help to solve the equation
$$y'=\frac{1 + y^2 - x^2}{2 x y}$$
Can I transform it into a homogeneous equation?
Actually this problem came from Apostol's Calculus I, 8.28.3
Find the ...
6
votes
2answers
71 views
I know this DE is solvable…
I need help with a seemingly simple looking diff equ
$$
x\frac {d^{2}y} {dx^{2}}+2y=0
$$
$$
\rightarrow \frac {d^{2}y} {dx^{2}}+2\frac {y} {x}=0
$$
$v= (\frac {y} {x})$ substitution isn't working ...
1
vote
2answers
37 views
Solve the following differential equation?
Please solve this with methods which a calc AB student can understand. $dx/dt-10x=(e^t)^4$. Find $x(t)$. If it's not clear, the RHS is e to the power of 4t. Thanks!
1
vote
1answer
29 views
Uniqueness of first order differential equation?
We have a theorem that says: Let $h: I\rightarrow \mathbb{R}$ and $g:J \rightarrow \mathbb{R}$ be continuous functions, $t_0 \in I $ and $y_0 \in \text{int(J)}$, then the differential equation ...
3
votes
1answer
31 views
solving a differential equation in form of integrals
I have given the differential equation $x'=\alpha(t)x+\beta(t)$.
How can you write the solution $x$ in form of integrals without $x$?
I've tried multiplying with the factor ...
1
vote
1answer
20 views
Functions defined by integrals.
I know what this says ( well sort of) they want me to take the derivative with respect to x i think? i may have to integrate with respect to y after i derivate before i can write the rest out. all n ...
0
votes
1answer
32 views
Finding the constant of integration in differential equations given conditions of $y$ and $x$
I am trying to solve a differential equation and find the constant of integration given the conditions that $y \to 0$ as $x \to \infty$. Given this condition I am unsure how to proceed on solving for ...
2
votes
1answer
37 views
Unique solution differential equation proof
Prove that there is a $\delta>0$ such that there is a unique solution of the differential equation $y'(t)=\sin(y(t))$ with $y(0)=1$ on the interval $[-\delta, \delta]$. How large can you choose ...
1
vote
1answer
65 views
Cancelling differentials
I'll start with an example.
In physics, $x(t)$ represents the $x$-position of a particle, and $v(t)$ its ($x$-)velocity. To determine the total displacement of a particle on the interval $[a, b]$, we ...
3
votes
2answers
34 views
Up and Down Motion (Two objects meeting in time?)
PROBLEM:
Suppose than an object is thrown upward with an initial velocity of 200ft/sec and that another one is thrown upward 5 seconds later with an initial velocity of 300ft/sec. When and where do ...
9
votes
0answers
101 views
Ramanujan style nested differential Equation
So I was exploring some math the other day... and I came across the following neat identity:
Given $y$ is a function of $x$ ($y(x)$) and
$$
y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + ...
2
votes
1answer
56 views
Solving for $x$ in this simple differential equation?
$\dfrac{dx}{dt}=2\dfrac{\sqrt{2g(\sin c- \sin x)}}{\sqrt{l}}$. $g$, $c$, and $l$ are all constants. Is there a way to solve for $x$ in terms of $t$ here? Once I did separation of variables and plugged ...
1
vote
3answers
92 views
Is this definite integral impossible?
From my understanding when you integrate $f(x)$ you get $F(x)+C$, and when finding a definite integral the $C's$ cancels out due to subtraction. However, I came across an example where the $C$ doesn't ...
1
vote
1answer
84 views
Partials and maximization
If we have that the contours of a response surface are elliptical and the response is given by the following function:
$$\large \exp\left(-\left(w^2 + \frac{1}{4}l^2 -\frac{1}{4} \cdot w \cdot ...
3
votes
1answer
61 views
Find the second derivative ${{{d^2}y} \over {d{x^2}}}$ in terms of t when $x = 3 - 2{t^2}$ and $y = {1 \over t}$
This is my attempt:
$\eqalign{
& x = 3 - 2{t^2} \cr
& y = {1 \over t} \cr
& {{dx} \over {dt}} = - 4t \cr
& {{dy} \over {dt}} = - {t^{ - 2}} = {{ - 1} \over {{t^2}}} ...
4
votes
1answer
105 views
Differential Equation$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{x^{2}+y^{2}}$
Please give me a hint to solve for $y$. Given,
$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{x^{2}+y^{2}}$$
I took $y=x\cdot\tan\theta$ but it was of no use.
1
vote
2answers
73 views
general solution and particular integral of $\dfrac{dy}{dx} + \dfrac{y \sin(x)}{1+\cos(x)} = \text{constant}$
I'm trying to find the general solution of this equation. I tried separation of variable but it does not work because we have $y$ collaborated with $x$ terms:
$$\dfrac{dy}{dx} + \dfrac{y ...
0
votes
1answer
27 views
Let $F(x,y,z)=\frac{-c r}{||r||^3}$ and $r = \langle x,y,z \rangle$
Compute $ \frac{ \partial F1}{\partial y}$ & $\frac{ \partial F2}{\partial x}$.
How do I do this if $F(x,y,z) = \frac{-cr}{||r||^3}$ is one function and not a vector of $<.F1.,.F2.>$?
0
votes
0answers
25 views
Truncation error and difference method
I am stuck on the following question. I am not sure of how to calculate the truncation error for the difference method
any help would be appreciated thank you!
0
votes
1answer
44 views
Modified Euler method
I am revising the modified euler method and would appreciate some help with this question:
The equation is $$y'=\frac{2}{x}y+x^2e^x, y(1)=0$$
Use modified euler method to calculate $y(1.1)$ taking ...
-2
votes
2answers
71 views
Ordinary differential equation question - help!
I've been learning the topic ordinary differential equations (first timer) and do not know how to solve this equation:
Consider the following ordinary differential equation
...
0
votes
2answers
87 views
Different methods and nonlinear systems
I'm trying to investigate nonlinear system numerical methods. So if we have a simple DE $x' = x$,
a) how to find the explicit solution $x(t)$ satisfying $x(0) = 1$?
b) how to use Euler's method to ...
0
votes
2answers
58 views
The general solution of the ODE $x^2y''+5xy'+13y=0$
I am trying to solve the ODE $x^2y''+5xy'+13y=0$.
Need a confirmation that answer is $$y=\dfrac{1}{x^2}\bigg(c_1 \cos(3\ln x)+c_2 \sin(3\ln x)\bigg)$$
I don't understand what quality standards are ...
1
vote
1answer
79 views
How does one find out the general solution of this second order differential equation?
I'm having trouble attacking this second order differential equation:
$$y''-tP(t)y'+P(t)y=0$$
Any help would be appreciated!
4
votes
2answers
44 views
Solving linear system of differential equations of 2nd order
I need to solve the following system of differential equations:
$$ \ddot{x} = 8x + 4y \\
\ddot{y} = -4x$$
Here's what I've done so far: I have reduced this system to a first order system, by saying ...
1
vote
0answers
40 views
Cohomologies of $\mathbb R^n$ with rational differential forms
We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
0
votes
0answers
45 views
Homogeneous ODE
I want to thank you for helping me with my homework. The question is:
Solve the homogeneous ordinary differential equation by matrix. Trial solution is: $λt$
...
0
votes
1answer
45 views
Calculus Rate of Change Problem
The rate of change of the mass $M(t)$ of a tree is approximated by $M'(t)=10t\sqrt{t+15}$ , where the mass is in grams and time $t$ is in days. At time $t =0$, the mass is 150,000 grams. Find the mass ...
2
votes
1answer
38 views
Orthogonality of eigenfunctions of a linear operator.
Suppose I have a linear operator
$$
\frac{\mathrm{d}^2}{\mathrm{d}r^2}+\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}
$$
and I want to find its eigenfunctions, that is, to solve the ODE
$$
...
1
vote
1answer
40 views
Simple differential equations not working?
So I was trying to solve $$\sin x+\cos x=1$$ I got $\tan x + 1=\sec x$. Now let $y=\tan x$. You have $$y+1=\sqrt{y'}$$ $$dy/dx=(y+1)^2$$ $$dy/(y+1)^2=dx$$ Now integrate both sides and you get ...
2
votes
1answer
50 views
Integration a function with a polynomial for a denominator
QUESTION
The following differential equation describes the amount of $x$ of KOH after time $t$:
$$\frac{dx}{dt} = k \left(n_1 - \frac{x}{2}\right)^2 \left(n_2 - \frac{x}{2}\right)^2 \left(n_3 - ...
1
vote
1answer
34 views
Solving differential equation for x
I have a field $\phi(x,t)=\sin(t+|x|)(\frac{x}{|x|})$ where x is a point vector and t is the current time.
If this field describes the acceleration of a particle at a point in space and time:
...
0
votes
1answer
52 views
Enigmatic optimization problem
My problem, which I proposed to myself months ago is based on the simple optimization problem
in which you find the best path for a lifeguard to rescue a drowning victim. Obviously the
shortest ...
0
votes
0answers
48 views
About calculating a Green's function
For a positive integer $d>=2$ and a real number $h_0$ I have the differential equation for a function $f$ of $x$,
$x^2 f'' + h_0^2 x f' = \frac{d(d-2)}{2} f$
The eigenfunctions are then $f \sim ...
2
votes
3answers
47 views
Solving $(f'(x))^2 = f(x)f''(x)$ with boundary conditions.
Let $f$ be a continuous real-valued function such that $$(f'(x))^2 = f(x)f''(x).$$ Suppose $f(0) = 1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.
I have this question in my book ...
0
votes
1answer
37 views
Differential equation solutions and change of variable
Consider the differential equation $$x''+\frac{a}{t}x'+\frac{b}{t^2}x=0$$ for $t>0$ where $a,b\in\mathbb{R}$ are constants.
I need to show that $x:\mathbb{R^+}\to\mathbb{R}$ is a solution of this ...
