Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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0answers
20 views

Diferential Equation (Separable Variables) “ODE”

Obtain the general solution (DE Separation of Variables) 1.) $y(5-x)dx = xydy$ 2.) $2ydx = 3xdy$ 3.) $y~e^{2x} dx = (4+e^{2x})dy$ 4.) $x \cos^2 ydx + \tan ydy=0$ 5.) $y'=y \sec x$
0
votes
0answers
17 views

Tricky Riccati ODE

I've come across the following Riccati DE from a problem I've been working on. $$y'(t)+\frac{g(t)(y(t))^2}{a}+\frac{x(t)(y(t))^2}{ah(t)+\frac{b}{y(t)}}=0$$ I tried using the substitution ...
2
votes
3answers
31 views

Use the Laplace transform to solve the initial value problem.

$$ y''-3y'+2y=e^{-t}; \quad\text{where}~ ~ y(2)=1, y'(2)=0 $$ Hint given: consider a translation of $y(x)$. I am stuck on this problem on our homework. I don't understand what they mean by a ...
0
votes
1answer
22 views

Why does the method of undetermined coefficients fails for exponential functions for in homogenous ODEs?

(By the "by the above method" it means the method of letting $y=ke^{rx}$ where $f(x)=e^{rx}$ in differential equations of the form: ) Now, I tried to confirm that the method fails when $r$ equals ...
0
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1answer
19 views

Do all Second Order ODEs (homogenous) have three solutions?

All differential equations of the form can be solved by the relevant quadratic equation that is their characteristics equation. However, although the quadratic provides two solutions, is it not true ...
0
votes
1answer
9 views

Why is the sum of two independent solutions of homogenous 2nd order ODEs a solution?

I am reading on the solutions of second order homogenous ODEs (linear), and came across this: Now, my questions is split into two parts: 1) I know that the independent solutions are equations, ...
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votes
0answers
44 views

What is the value of $x$ in $\frac{x}{x}$=$\frac{3}{3}$? [on hold]

Could it be true to say, taking the numerators or denominators $x=3$ like in differential equations.
1
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0answers
22 views

Solving Ricatti equation $v'=av^2+bv+c$

pretty much as the title says. I am trying to solve Ricatti equation $$ v(t)'=av(t)^2+bv(t)+c $$ where $a$,$b$,$c$ are real-valued constants. Wolfram|Alpha gives the solution (here), but I am not ...
0
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0answers
35 views

{$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$

I have a question that it maybe simple but I can not understand why we have : {$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ ...
4
votes
1answer
62 views

Problem with Justifying the Formula for First Order Seperable Differential Equations

I am reading this text http://www.math-cs.gordon.edu/courses/ma225/handouts/sepvar.pdf to justify the method to solve first order seperable differentiable equations, where we are told first told that: ...
3
votes
1answer
55 views

transform integral to differential equations

I found a similar system of integral equations in a paper. It says that it can be solved by differentiating and then using standard techniques. My question is, how can I differentiate such a system in ...
1
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0answers
13 views

Recognizing and using hypergeometric function

Some expressions that interest me end up having something to do with hyper geometric function. I want to be able to derive such results myself. Where do I begin? For example, the equation $$ ...
1
vote
0answers
13 views

Solving a system of ODEs

I have three ordinary differential equations, with 3 dependent variables, $n_i$, $n_e$ and $\phi$, as follows: $$\nabla (-\frac{dn_i}{dx}-k_1n_i\frac{d\phi}{dx})=0$$ $$\nabla ...
3
votes
1answer
56 views

The real equation of a pendulum

In physics I never solve the equation $\ddot\theta = \sin(\theta)$. Instead, we used the approximation $\theta = \sin(\theta)$ for small angles and then it was easy to solve. I didn't do any physics ...
1
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0answers
40 views

A system of partial differential equations

I have 6 partial differential equations that in the first look they don't seem very difficult, but all my efforts for solving them were unsuccessful. $$\frac{\partial f(x,y,z)}{\partial ...
2
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0answers
29 views

Finding a Lyapunov Function for a system involving a trigonometric function

I'm dealing with determining if $(0,0)$ is stable or not for the following system via constructing a Lyapunov function. The system is $$ \begin{cases} x'(t)=(1-x)y+x^2\sin{(x)}& \\ ...
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votes
0answers
24 views

Find the population [on hold]

Every year, the emigration rate from country A to B is 𝛼 (0 < 𝛼 < 1), whereas the emigration rate from country B to A is 𝛽 (0 < 𝛽 < 1). Note that the fluctuation in population of both ...
0
votes
3answers
45 views

differential equation with substituion

Solve for y: $y'\tan(x+y)=1-\tan(x+y)$ so far I have made the substituion $u=x+y$, which yields $\frac{du}{dx}=1+\frac{dy}{dx}$. However, I am not sure what to do from here.
0
votes
2answers
24 views

Is it possible to say that $L(f^n)=s^nL(f)$ when the differential equation is not in the rest condition?

Question Use the Laplace transform to solve the following equation: $y'+2y=\cos(3t)$ ; where $y(0)=1$ In class our teacher wrote that "When in rest condition: $L(f^n)=s^nL(f)$", but I want to use ...
1
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3answers
39 views

Find the inverse Laplace transform of: $\frac{1}{(s^2+a^2)(s^2+b^2)}$

I'm having trouble doing this homework problem because I'm not sure how to deal with the $a$ and $b$. I did it the usual way we were taught - use partial fraction decomposition and then try to solve ...
2
votes
1answer
68 views
+50

Non linear second order ODE up to $O(\epsilon) $ for $v_{xx}-\left(v^{3}-v\right)-\varepsilon\frac{1}{2}\left(1-v^{2}\right)=0$

I really need help solving this : $$v_{xx}-\left(v^{3}-v\right)-\varepsilon\frac{1}{2}\left(1-v^{2}\right)=0 $$ With boundary conditions : $$ v(\pm \infty )=-1+\frac{1}{4}\epsilon $$ I need to ...
0
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0answers
29 views

Taylor Series Methods with Quadrature. Local Truncation Error [on hold]

I don't know where to begin with this question. Advice would be helpful. Suppose that a differential equation is solved numerically on an interval $[a,b]$ and that the local truncation error is ...
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votes
3answers
39 views

Differentiation with respect to a constant variable? [on hold]

Let $y=f(x)$. If we are trying to find $f^{\prime}(x)$ and we know that in the domain we are trying to find $f^{\prime}(x)$ in, $x$ is constant , then what is $f^{\prime}(x)$? Is it zero?
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votes
4answers
99 views

If $f(x)\ll1$ is it safe to assume that $f^{\prime}(x)\ll1$?

If $$f(x)\ll1$$ is it safe to assume that $$f^{\prime}(x)\ll1$$
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0answers
22 views

Proving one single solution for an ODE

let $f(y)$ be a continuous function in $R$. $f(y)=0$ only for $y=y_0$. as a result the integral: $\int_y^{y_0} \frac1{f(x)} dx $ diverges for every $y$. prove that the following problem: ...
2
votes
1answer
37 views

Wronskian of two differential equation solutions

Let $f$ and $g$ be the solutions of the homogeneous linear equation: $$y'' + p(x)y' + q(x)y = 0$$ and $p(x)$ and $q(x)$ are continuous in segment $I$. Is it true, that if the wronskian of $f$ and ...
2
votes
2answers
34 views

Differentiation with dependent variable

Let $$ F(x, y) = x^3 + 7 y^2 x^4 - (2 x - y)^3 $$ and let $y=f(x)=x^2+1$. Is it correct to write $$ \frac{\partial F}{\partial y}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial y}? $$ ...
6
votes
1answer
52 views

Are there closed curves for which acceleration is orthogonal to position?

Can we find $\vec{f} : \mathbb{R}\rightarrow \mathbb{R}^3 $ such that $\vec{f}(t) \cdot \frac{d^2 \vec{f}(t)}{dt^2} =0$ and $\vec{f}(0) = \vec{f}(T)$ for some $T >0$ ? Exclude the trivial cases. I ...
0
votes
2answers
54 views

Solve the following initial value problem: $2y''+y'-y=e^{3t}$

$$ 2y''+y'-y=e^{3t}; \text{ with } y(0)=2,\ y'(0)=0 $$ I got to this point: $$ L(y)=\frac{1}{(s-3)^2}\cdot\frac{1}{(2s-1)(s+1)} $$ but now I'm not sure what to do with these polynomials. I know ...
0
votes
1answer
28 views

solving equation in terms of $w_1$ and $w_2$

I have a a physics problem involves the following equation $$\tan(\alpha) = \frac{(w_1 + w_2)^{1/2}}{w_3}$$ from a certain set of equations that I use I derive the following equation: ...
0
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1answer
31 views

Why can we subtract two terms and use this as a way to simplify the overall expression?

Our professor was doing a Laplacian transform example in class. Original problem: $$ y''+4y=\sin t $$ He was working on the problem and got to this step: $$ ...
0
votes
1answer
14 views

ODE boundary condition and integer values?

When separating the variables of the 3d wave equation we generate (amongst others) the following ODE. \begin{equation} \frac{d^2\Phi}{d\phi^2}=-m^2\Phi \end{equation} Where $m$ is the separation ...
1
vote
1answer
26 views

ODE, Picard approximation of a second order equation: How do I make sure that this is correct.

I have the following problem: $$\ddot{x} + \dot{x}^2-2x=0$$ and I.V are: $x(0)=1 \qquad$ $\dot{x}(0) = 0$. and I need to find two first "Picard" approximations. I first arranged it in the form ...
0
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3answers
110 views

Books various maths subjects [on hold]

I am a Civil Engineering student and i am planning on following physics in my career.I want to be ready for the advanced undergraduate courses that i will attend to,so i need to learn Differential ...
1
vote
2answers
55 views

Practical use for negative $dt.$

I am writing a section of notes for Calculus 1 on related rates. In the section where I discuss differentials, I write that the quantity $dt$ must be nonnegative. I imagined the only reason it would ...
2
votes
2answers
40 views

Finding the kernel, eigenvalues, and eigenvectors of the operator $L(x) := x'' + 3 x' + 4 x$

I want to find the kernel, eigenvalues and eigenvectors of the differential operator: $$L(x)=x''+3x'-4x$$ on the $\Bbb C \space \space \text{vectorspace} \space \space C^{\infty}(\Bbb R)$ as well ...
0
votes
1answer
25 views

Differential Equation with a Motorboat

A motorboat and its load weigh 2150N. Assuming the propeller force is constant and equal to 110 newtons and water resistance is equal numerically to 6.7V Newton where V is the velocity at any instant ...
2
votes
2answers
37 views

stability of equilibria for $n$-dimensional nonlinear systems of differential equations: examples

I'm currently self-studying dynamical systems. I'm trying to summarize what can be said about the stability of equilibrium points for an $n$-dimensional non-linear system of differential equations: ...
1
vote
1answer
28 views

How close should be the boundary value of $x$ and $y$ to ensure that $|x(t)-y(t)|<0.1$

I was given the following differential equation: $$y' = \sin y\cdot \sin t+y\cos t$$ Say that $x(t)$ and $y(t)$ solve this equation, and that $x(t_0) = x_0$ , $y(t_0)=y_0$. Find $\varepsilon$ small ...
0
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0answers
22 views

Name of numerical methods for second-order differential equation

Numerical methods that try to solve first-order differential equations of the form: $$ \frac{\partial}{\partial t} y = f(y,t) $$ are often Runge-Kutta methods, and there is a whole family of ...
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0answers
17 views

instability of a differential equation

I have the following simple differential equation ; $$\frac{dz}{dt}=\left(\alpha_{1}+\alpha_{2}\right)-q_{t}\left(z_{t}-1\right)$$ I know that $\alpha_{1}$ and $\alpha_{2}$ are positive constants. ...
0
votes
1answer
12 views

What's the difference between wave equation in PDE form and wave equation in normal form?

What's the difference between "wave equation in partial derivative form" and "wave equation in y(x,t) form" ? Are they both same? And why "wave equation in in y(x,t) form" is the solution of "wave ...
0
votes
1answer
48 views

Differential equations, chemical reactions

A chemical substance A changes into substance B at rate $\alpha$ times the amount of A present. Substance B changes into C at rate $\beta$ times the amount of B present. If initially only substance A ...
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vote
3answers
148 views

Transforming to a homogeneous equation

Consider the equation $$\frac{dy}{dx}=F(\frac{ax+by+c}{dx+ey+f})$$ Show that if $ae \neq bd$ then there exists constants $h \; , \; k$ such that the substitution $x=z-h$ and $y=w-k$ converts the ...
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0answers
29 views

Finding differential equation satisfied by the following families of curves [on hold]

How to find differential equation satisfied: $$y=e^{Cx}$$
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3answers
50 views

How to solve the following problem?

How to solve the following ODE? $$y′ − y = 2x − 3;\ y(0) = 1$$
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2answers
41 views

Finding solutions to the ODE [on hold]

Please help to find general and particular solutions which satisfy the given additional condition: $y′ = e^{x+y};\ y′(1) = 1$
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1answer
25 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
0
votes
2answers
23 views

ordinary differential equations-morphogen gradient

I am reading a paper by Merkin and Sleeman (2005) Find the approximation solution of $(u')^2=\frac{2}{k}(u-\frac{1}{k}\ln(1+ku)); ~~u(0)=1$ for $k$ sufficiently small. they gave the following ...
1
vote
0answers
48 views

Two ODEs, why is one solution the solution of the other?

Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ Caratheodory, and we know that if it has a solution, it ...