1
vote
1answer
27 views

Meaning of $ dx \times dy = k $

Does $ dx \times dy = k $ have a mathematical meaning? What about when considering $y = y(x)$?
0
votes
0answers
13 views

predictor-corrector method and stability

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
0
votes
0answers
8 views

Finite difference scheme and its stability

The Finite difference scheme: \begin{equation} y_{n+3}-y_{n+1}= \frac {h}{3}(f_{n}-2f_{n+1}+7f_{n+2}) \end{equation} Deduce that the scheme is convergent and find its interval of absolute stability(if ...
0
votes
0answers
5 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
1
vote
1answer
28 views

Solving a differential equation?

I'm trying to analyze the transient state of a RC circuit. My book gives me the following differential equation: $$\frac{d(v(t))}{dt} + av(t) = c$$ for some constants $a$ and $c$. The book thens ...
0
votes
0answers
45 views

Demonstrate identity $\int_{0}^{x} \int_{0}^{\xi}f(s)dsd\xi=\int_{0}^{x}(x-\xi)f(\xi)d\xi$

I have trouble doing the following problem, I have not been able to make even the first part, I was hoping someone could help me. The problem is: Show that: $$\int_{0}^{x} ...
1
vote
3answers
34 views

1st order differential equation

I am given the following: $$ \begin{cases} x \ln x \frac{dy}{dx}+y + x = 0, &\mbox{if}\quad x>1, \\ y = 0, &\mbox{if} \quad x=e \end{cases} $$ I tried to separate it and got this: $$ -y \ ...
1
vote
2answers
37 views

How to solve $y''+9y=-18\sin{3x}-18e^{3x}$?

Here is my solution so far: $$y''+9y=-18\sin{3x}-18e^{3x}$$ 1.Find complementary soultion.$$y''+9y=0$$ assuming that solution will be in form $e^{kx}$, substitute $y=e^{kx}$, $$k^2e^{kx}+9e^{kx}=0$$ ...
2
votes
0answers
30 views

Solve by separating variables

$$\frac{dy}{dt}=e^y +1$$ I've tried: $$dy/dt - e^y = 1 $$ $$\Leftrightarrow y' - e^y dt = 1 dt$$ But I'm not sure what to do next or if I'm even doing this right!
1
vote
1answer
24 views

Solving first order differential equation

I am given this: $$(2x+1)\frac{dy}{dx}+y = 0$$ I tried this: $$\frac{1}{(2x+1)} dx = \frac{-1}{y} dy$$ Then integrated the above sum and got this: $$ \frac{ln(2x+1)}{2}= -ln(y)$$ The answer is: ...
0
votes
0answers
20 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
0
votes
1answer
37 views

What is the best approach to solve $ 4y^3 y''=16 y^4 -1$?

How can I solve this DE: $$ 4y^3 y''=16 y^4 -1$$ I really would not bother asking if Wolfram alpha had not exceeded comp. time and not shown me step-by-step solution.
0
votes
1answer
24 views

How to solve $xy'=2\sqrt{x^2+y^2}+y$?

How to solve: $$xy'=2\sqrt{x^2+y^2}+y$$ And what would be the standard form to illustrate this situation? (e.g. $y' +P(x)y=Q(x)$ would be standard form of first order linear differential equation)
1
vote
0answers
28 views

Total differentiation

For each of the functions below use the total diferential to approximate the change in $Y$ due to the given changes in $X$ and $Z$: $Y= X^2 + 4X -Z^2 -2XZ$, where $X=1$ and $Z = 4$ , and $\Delta X=2$ ...
1
vote
0answers
23 views

Predictor-Corrector for Adams-Moulton

What is the order of the corrector of Adams-Moulton type required in order to apply Milne's method for estimating the error in PECE mode? Find the coefficient of the leading term in the truncation ...
4
votes
1answer
31 views

Unsure with second order complex differential equations

Solve $$y'' - 4y' + 5y = 0 $$ Where $y(0) = 0 \ , \ y'(0) = 2$. So I solve this as a second degree polynomial (no idea why) $$\frac{4 \pm \sqrt{16-20}}{2} = 2 \pm 2i$$ So the CASE III solution as ...
3
votes
1answer
82 views

Is there any general function $x(t)$ that gives the solution to $x''(t) = k/x(t)^2$, where k is a constant?

In physics class, I often come across various inverse square law equations like the following: $F_G= G\frac{m_1m_2}{r^2}$ $F_E = k_e\frac{q_1q_2}{r^2}$ Specifically, we are typically given ...
0
votes
3answers
33 views

First order ODE problem

Solve $$2y'+3y = 0$$ So my integrating factor $p(x) = 3$. So I multiply both sides by $$e^{\int 3 \ dx}$$ And get $$e^{3x} (2y'+3y)=0$$ I now have to integrate both sides but the trick is that I ...
0
votes
3answers
34 views

Solving differential equation (equation whose unknown is a function)

I had homework asked us to solve a differential equation I did it myself but now I'm stuck i found: $$\int\frac1ydy=6x$$ How can I continue?
18
votes
2answers
224 views

When does $(uv)'=u'v'?$

In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this dream product rule true? There are ...
1
vote
1answer
48 views

2nd order linear differential equation

Attempt: a) oscillating solutions will occur for $\alpha^2 < 4\beta$, no oscillation if $\alpha^2 > 4\beta$ with $\beta > 0$. (is this necessary?) for some $\sigma$, we have ...
2
votes
2answers
51 views

Procedure for 3 by 3 Non homogenous Linear systems (Differential Equations)

Here is the problem I have. $$x^{'}(t)=Ax(t)+f(t)$$ where $A=\begin{pmatrix} 5&-3&-2\\8&-5&-4\\-4&3&3 \end{pmatrix} f(t)=\begin{pmatrix} -\sin (t)\\ 0 \\ 2 \end{pmatrix}$ I am ...
1
vote
1answer
87 views

Integrate $\int^{ln(2)}_0 (3e^u - e^{2u} - 2)\sin(nu)du$

I'm having trouble integrating this function $$\begin{equation} \begin{split} f(x) & = \int^1_0x(1-x)\sqrt{1+x}\sqrt{1+x}\sin(n \ln(1+x))/[(1+x)^2] = \\ & = ...
0
votes
1answer
40 views

Verify the given function including the integral $e^{-x^2}$

I'm really stuck trying to verify that the given function is a solution of the differential equation. I've attempted applying converting it to polar coordinates but I don't think I'm on the right ...
1
vote
2answers
75 views

Solutions for an ODE

I am looking for a solution of the ODE $x'(t)=x(t)+\frac{1}{1+e^{t}}$ which has finite limit when $t\rightarrow \infty$, I already find that the solutions are $e^t \ln(1+e^t)-te^t-1$ however these ...
0
votes
1answer
57 views

Find an equation of the curve that passes through the point (0, 1) and whose slope at (x, y) is $xe^y$ [closed]

Using differential equations and first order separable equations Find an equation of the curve that passes through the point $(0, 1)$ and whose slope at $(x, y)$ is $xe^y$
1
vote
0answers
39 views

Show that $y/x$ tends to a finite limit as $x \to + \infty$ and determine this limit.

Let $y=f(x)$ be that solution of the differential equation $$y' = \frac{2y^2+x}{3y^2+5}$$ which satisfies the initial condition $f(0)=0$. (Do not attempt to solve this differential equation.) (a) ...
1
vote
1answer
25 views

problem about population growth

At the beginning of the Gold Rush, the population of Coyote Gulch,Arizona was $365$.From then on ,the population would have grown by a factor of $e$ each year,except for the high rate of ...
1
vote
2answers
63 views

Legendre trigonometric form

Consider the Legendre equation for a function $y(x)$ defined in the interval $-1 < x < 1$ By a change of variable $x = cos \theta$ derive the trigonometric form of Legendre equation for a ...
1
vote
2answers
39 views

Help with separable differential equation? $\frac{dy}{dx} =2y^2$

I'm new to separable differential equations, and I'm stuck on this question: $\frac{dy}{dx} =2y^2$ Using the initial condition $y(2)=3$, find $y(1)$. So far I've integrated to get $\frac{dy}{dx} ...
0
votes
0answers
41 views

Using Picard-Lindelöf Theorem to elegantly demonstrate uniqueness of an IVP

I am trying to keep this question clean and short, therefore I won't write down the entire theorem of Picard-Lindelöf here. Problem: $$y'=1+y^2 =:F(y), \ y(0)=0 $$ Find a solution on a ...
0
votes
4answers
65 views

Differential Equation: $f'(x) = f(x) (1-f(x))$

I'm lost on the following problem: Find the function f(x) such that f'(x) = f(x)(1-f(x)) and f(0) = 1/7. (Use f for f(x) in your equation). I'm assuming I can write this as: $$ \frac {df}{dx} = ...
-3
votes
0answers
30 views

Differential equations problem, help!? [duplicate]

If I have an equation $y''+ay'+by=f(x)$ where $y=h(x)$ and $z'''+az''+bz''=f '(x)$ is it correct to say that $z$ must take the form $h(x)+ \text{const}$ ?
0
votes
1answer
43 views

Derivative of a differential equation help??

Please can someone explain this to me in detail: if $y''+4y'+3y=14\cos(2x)$ and $z'''+4z''+3z'=-28\sin(2x)$ show that the $z=y+c$ where $c$ is a constant I know the second is the integral of the first ...
1
vote
1answer
42 views

Finding a Differential Equation that Satisfies an Initial Condition

Find the solution of the differential equation that satisfies the given initial condition: $$ \frac{dL}{dt} = kL^2ln(t), L(1) = -8$$ The thing that's really screwing me up here is that darn k. I've ...
0
votes
2answers
34 views

Solve the separable differential equation

This question is really basic but I'm having a hard time figuring it out.. Any help is appreciated. $$y' = 6y^2$$ Using the initial condition: $y(6) = 3$, find $y(1)$. I tried this by ...
1
vote
1answer
194 views

calculus, predator-prey system

The following system describes a predator prey system in which the prey has an Allee effect. What is the threshold of the prey to persist when alone? Find the nullclines and the steady states of the ...
2
votes
0answers
48 views

Second order nonlinear ordinary differential equation. Help please

Can someone help me with this differential equation $$ay''(t)y(t)+2y'(t)=\left(b+\frac{c}{t^2}\right)y(t)^2$$
1
vote
1answer
28 views

Approximation of the solution of an IVP

Consider the initial value problem $$\frac{dy}{dx} = x^2 + y^2, \\ y(0) = 0$$ on D = {|x| <= 1, |y| <= 1} Find the third approximation to the solution If someone could maybe walk me through ...
2
votes
2answers
75 views

How to solve this Diff Eq (with multiple terms)

$$\frac{dy}{dx}=7xy$$ I know this turns into $$\frac{dy}{y}=7xdx$$ .....etc. But, how do you solve the following: $$\frac{dy}{dx}=7x+y$$ Not sure how to seperate the parts to respective sides.
0
votes
1answer
32 views

First order ODE with $f'(x) = 810(10)^x$

I'm trying to find an explicit form of the series $f(0) = 89.1,f(1) = 899.1,f(2) = 8999.1, \cdots$. My first though was to take the derivative and integrate it, which I've done before with a fair ...
1
vote
1answer
22 views

Diff EQ. Problem (Eigenvector issue)

Find the general solution of $\textbf{x}^{'}=\begin{pmatrix} -1&-4\\1&-1\end{pmatrix}\textbf{x}$. The eigenvalues I found are $-1 \pm 2i $ and I chose $-1-2i$ to be my eigenvector and found ...
1
vote
1answer
21 views

General Solution to a Differential EQ with complex eigenvalues.

I need a little explanation here the general solution is $$x(t)=c_1u(t)+c_2v(t)$$ where $u(t)=e^{\lambda t}(\textbf{a} \cos \mu t-\textbf{b} \sin \mu t$ and $v(t)=e^{\lambda t}(\textbf{a} \sin \mu t ...
0
votes
1answer
24 views

How do I find the required acceleration to equalize moving objects with different start velocities?

If I have different objects that start moving at one direction with different velocities, how do i find how much I must accelerate each, so they soon have almost identical velocity?
2
votes
2answers
35 views

Reducing to first-order differential systems

Hello, in order to do this, I am aware I need to substitute $x''$ for say $a'$ and $y''$ for say $b'$, but I'm unsure of how this will yield 4 equations? Furthermore, in terms of wording of the ...
0
votes
2answers
21 views

LaPlace transform of the delta function

I am having difficulty taking the laplace transform of $$\delta(t-2\pi)\cos(t)$$ I know that if we have the delta function it is just $e^{-cs}$ but what about the product?
1
vote
1answer
27 views

Finding the General Solution to the system of equation

Find the General solution of $\textbf{x}^{'}=\begin{pmatrix} 2&2+i\\-1&-1-i\\ \end{pmatrix}\textbf{x}$ I started out by finding the eigenvalues. ...
1
vote
0answers
46 views

Number of zeros of Wronskian

Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & ...
1
vote
1answer
47 views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
11
votes
2answers
162 views

If $f(x) + f'(x) + f''(x) \to A$ as $x \to \infty$ then show that $f(x) \to A$ as $x \to \infty$

This problem is an extension to the simpler problem which deals with $f(x) + f'(x) \to A$ as $x \to \infty$ (see problem 2 on my blog). If $f$ is twice continuously differentiable in some interval ...