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

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1answer
40 views

Differential equation, hint

$$x^2y'^2-2(xy-2)y' +y^2=0$$ I have tried to determine y' using x, y $$y'=-\frac{{\sqrt{1-xy}-1}^2}{x^2}$$ And I don't know what to do next.
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2answers
43 views

Homogenous Differential Equation

The problem is to solve for the general solution of: $$yy' + x = (x^2 + y^2)^{1/2}$$ My teacher said to homogenize the equation as a hint during our exam, and afterwards in the solutions he gave ...
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0answers
27 views

Using Frobenius' Theorem for 3 functions in 2 variables [closed]

i) 1) $v= \frac{\partial u}{\partial x}$ 2) $w= \frac{\partial u}{\partial t}$ 3) $\frac{\partial v}{\partial t}= \frac{\partial w}{\partial x}$ 4) $\frac{\partial v}{\partial x}= \frac{\partial ...
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0answers
12 views

A change of form in harmonic motion

Let $x(t)=$ $c_1\cos\omega_0t+c_2\sin\omega_0t$ , $c_1=c\cdot \cos\delta$, $c_2=-c\cdot \sin\delta$, and $c>0$, can we say by this change of form $c_1$ and $c_2$ still take value on all real ...
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0answers
21 views

Using Frobenius Theorem to solve a fairly general system of 1st order PDEs

It has been proved that if $\mathbb{X}$ and $\mathbb{Y}$ be the vector fields on $\mathbb{R}^3$ given by $\mathbb{X}(x,y,z)=(1,0,p(x,y)r(z))$ $\mathbb{Y}(x,y,z)=(0,1,q(x,y)r(z))$ ...
0
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1answer
38 views

solving Differential Equation (hint)

$$(yy')^3 = 27x(y^2-2x^2)$$ I tried a lot, but one what i see is $yy'=(y^2)'$ and then we get $z'^3 = 216x(z-2x^2)$ I have no idea, please, hint a substitution.
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1answer
72 views

Problem integrating in problem using the Poincaré Lemma

a) It is easy to show that $d\beta=0$. b) $\begin{align}\hat{\mathbb{X}}_t &= (\frac{\partial}{\partial t}\hat{\Phi}_t) \hat{\Phi}_t^{-1}) \\ &= (\frac{\partial}{\partial t}\hat{\Phi}_t) ...
1
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1answer
32 views

Differential equation, for which values of 'a' does this have a bounded solution?

Let $f(t) = f(t$) be the 2pi periodic("sawtooth wave"), f(t) = t for $0 \leq t \leq 2\pi$ and consider the equation $$y^{\prime \prime} + a^2y = f$$ For which values of $a$ (here $a$ >0) does this ...
1
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1answer
62 views

Differential Equation help: $\frac{dy}{dx}=\frac{y-3}{x^2 +y^2}$

The question is: solve for $y,$ $\dfrac{dy}{dx}=\dfrac{y-3}{x^2 +y^2}$ given it passes through $(0,1)$. I am struggling to find a way to separate the variables. Also as a side question, if you have ...
1
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0answers
88 views

how to solve Schrödinger equation

I would like to solve a complete solutions of the Schrödinger equation for a particle with time & position dependent mass ($m(x,t)$) moving in a potential $V(x,t)$. Any suggestions to solve ...
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2answers
22 views

Factoring Quadratic

I have used the substitution P = dy/dx to solve a first-order D.E of degree 4, so I got this: I have to show that the above statement can be written as: I tried to factor out first by taking p a ...
4
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1answer
106 views

Solving a PDE with Feynman-Kac Formula

I'm trying to solve this PDE using Feynman-Kac formula Now i follow the regular steps Here is where I don't know how to proceed. How do I calculate this expectation?
2
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2answers
50 views

Separable equation

I had $y=e^{4\ln|x|}+e^{4C}$, then simplified to $y=e^4 \cdot e^{\ln|x|} +e^{4C}=A\ln|x|+C_2$. This seems to be wrong and should've been $e^{\ln|x^4|}+e^{4C}$. Why is what I initially did wrong? ...
0
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1answer
32 views

Does the reduction of order yield the general solution of a differential equation?

Given a ODE of order 2, $$c_0(x) y''+c_1(x) y'+ c_2(x) y=q(x)$$ if we somehow find a solution $h(x)$ for homogeneous equation, then we can reduce the order of the non-homogeneous equation by letting ...
1
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0answers
35 views

Reducing a system of differential equations

Let $\mathbf F$ be a system of 1st order differential equations in $n>3$ variables $$\mathbf{F} : \mathbb{R}^n \to \mathbb{R}^n$$ $$\frac{d\mathbf{u}}{dt} = \mathbf{F}(\mathbf{u})$$ such that ...
2
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2answers
31 views

Inverse Laplace transform shifting error

I am doing the inverse Laplace transform of the function: $\frac{e^{-s}}{s-1}$. I am solving and receiving the answer: $e^t\mathcal{U}(t-1)$, however the correct answer is $e^{t-1}\mathcal{U}(t-1).$ ...
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0answers
35 views

directional derivatives and gradient

It is raining and rainwater is running off an ellipsoidal dome with equation 4 x^ 2 + y ^2 + 4 z^ 2 = 16, where z ≥ 0. Given that gravity will cause the raindrops to slide down the dome as rapidly as ...
0
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1answer
25 views

1st order differential equation with x(t) and y(t) in one equation

$y$ and $x$ are all in term of $t$ but after I have found the integrating factor and multiply is to the both side, then RHS will become $xe^{Rt/L}$ and don't know how to continue integrating the ...
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0answers
41 views

How do you do this Laplace transform?

How do you do the Laplace transform of the following equation? I know there's a trig identity in here somewhere, but I've got no idea how to do/use it. $$\mathcal{L}\lbrace\cos{4t}\cos{2t}\rbrace$$
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0answers
24 views

Find the frequencies that produce resonance in system of second order differential equations [closed]

The Eigen values are lamda= -5 and -2. A. What frequencies (w) would produce resonance in the system of second order differential equations x1''=-3x1 + 2x2 + coswt x2''=x1 - 4x2 + cos wt B. find ...
3
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1answer
100 views

Describing non-vanishing $1$-forms on two dimensional manifolds.

Let $h_1 \mathrm{d}x_1 + h_2 \mathrm{d}x_2$ be a non-vanishing $1$-form on a $2$-dimensional manifold. Why do locally exist smooth functions $f,g$ with $f\mathrm{d}g= h_1 \mathrm{d}x_1 + h_2 ...
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0answers
92 views

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
1
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1answer
56 views

ordinary differential equation solving

I have a diffeq: I have a nonlinear Diffeq: $$\frac{d^2x}{dt^2}+\beta \frac{dx}{dt}+\varepsilon e^{- \lambda x} = f(t) $$ where $f(t)$ is a function that is known, and $\beta$ and $\lambda$ are ...
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2answers
111 views

What is a good text which introduces ODE in a very general setting?

For your information, I have studied almost all undergraduate mathematics except for differential equation and I'm really comfortable with what I have learned. Moreover, I have taken an one semester ...
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0answers
9 views

method of characteristics in a nutshell

Good morning everybody, I need a quick reference for the following inhomogeneous first-order pde... namely $$f(x,y,z)=A\partial_x\varphi+B\partial_y\varphi+C\partial_z\varphi,$$ where $\varphi\in ...
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1answer
26 views

Diffusion-Reaction PDE - radial coordinate

I am trying to obtain an expression for the concentration $C$ based on this stationary equation : $\frac{\partial C}{\partial t} = \frac{1}{r} \frac{d}{dr} \left(r \frac{\partial C}{\partial ...
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0answers
16 views

Optimal time control for the system of two non-linear ODE

I have the following system of two non-linear ODE with one control variable (modified model of Lotka-Volterra): Here is $\alpha, \beta, \gamma, \delta$ - some constants, $u$ - control variable. ...
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0answers
24 views

Solution existence on $ax + by = c$

I have to produce an program which resolve the following equation: $ax + by = c$ With the following condition: $a$, $b$ and $c$ are known positive integer. $x$ and $y$ are positive ...
3
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0answers
23 views

when does a partial differential equation have unique solution?

The differential equation $ xu_x + yu_y = 2u$ satisfying the initial conditions $y = xg(x), u=f(x)$ with $f(x) = 2x, g(x) = 1$, has no solution $f(x) = 2x^2, g(x) =1$, has infinite number of ...
2
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0answers
13 views

Identifying Hamiltonian Systems with Phase Portrait

the following is a homework question (that isn't going to be graded) and I'm not sure how to do it. I know that the solution trajectories cannot cross the H(x,y)=constant curves, but I'm not sure ...
0
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2answers
23 views

Linear differential equation with fundamental solutions $\sin x$ and $e^x$

How could I just make a differential equation out of the blue that has the fundamental solutions $y=c_1\sin x$ and $y=c_2e^x$.
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0answers
13 views

Do these “algebraically well behaved” Function Spaces, exist?

Do there exist any Sets of Functions which are some combination of: Algebraically Closed: In the sense of Algebraic Functions. Differentially Closed: In the sense of Differentially Closed Fields. ...
1
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1answer
21 views

find the nullclines

I don't know how to find the nullclines in the system (see picture attached) according to my professor (0,0) is one of them. I'm so confuse looking for it.
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0answers
15 views

show that the bifurcation occur at an exact point

Hi I'm having trouble with this problem. I have everything completed, but not the parte where I have to show that the saddle node bifurcation occur at k= -a + 1/2(a)^(1/2) and k= -a - 1/2(a)^(1/2). I ...
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0answers
14 views

Solution of a differential equation with problem of Cauchy

The question is the next: What can I say from the existence, uniqueness and continuos dependence of the solution? Is this a strongly continuos one-parameter group or a semigroup. $ \left\{ ...
2
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1answer
25 views

Why can $1 + \cos(t+y) + \cos(t+y)(\frac{dy}{dt}) = 0$ be written in the form $(\frac{d}{dt})\lbrack t + \sin(t+y) \rbrack = 0$?

I'm reading Differential Equations and Their Applications by Martin Braun. In subsection 1.9, which deals with exact equations, the author writes: Example 1. The equation $1 + \cos(t+y) + ...
0
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1answer
16 views

DE Power Series solution centered at $0$ but DE not defined at $0$.

I took a test in which I was asked to give a solution in terms of a power series for the equation: $$x^2y''(x)-xy'(x)+(1-x)y(x)=0,~~~~~~~x>0.$$ At first I began to work on a power series centered ...
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0answers
25 views

Questio on general differential equations

Let $f$, $g$ be functions. If $f+g$, $f+g+1$, $f+2g$ are solution to $$L[y]=y'' + p(x)y' +(q(x)-x^2)y=x,$$ where $x>0$. Solve $L[y]=x$, $x>0$ using variation of parameters undetermined ...
1
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1answer
17 views

A sufficient condition that domain of solution of differential equation became $\mathbb R$

If $ f:\mathbb R^n\to \mathbb R $ be bounded and continous then differential equation $$x'=f(x)$$ has a solution with domain $\mathbb R$. outlook of proof : if the maximal domain of solution is ...
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1answer
39 views

Tough second order differential equation2

I have asked similar question before but it turns out that the $E_0$ depends on (r,z). It makes the solution complicated. Any comments appreciated. ...
1
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1answer
36 views

Differential equation

$$x^2(x^2+1)y''-2x^3y'+2(x^2-1)y=0$$ Anyway my tutor assumes that one of the solutions must be $y=x^2$, yet I can't seem to prove this, I tried using $y=x^n$ but I can't seem to get an answer that ...
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1answer
46 views

How would I solve this ODE?

My ODE looks as follows: $(f(x))^2 + f(x)f'(x) x = c$ Any ideas if or how I could solve this? I do know how to solve it for c=0 but Im not sure what the idea is for c $\neq$ 0 Thanks.
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1answer
18 views

Dynamical Systems, a question about first order DE asking for an example. [closed]

Construct an example of a differential equation depending on a parameter a for which some solutions do not depend continuously on a.
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1answer
42 views

Modelling population with $\frac{dP}{dt}=P(\beta - \delta P)$

The population $P(t)$ of a biological species can be modelled by $$ \frac{dP}{dt}=P(\beta - \delta P) $$ subject to $P(0)=P_0$ where $\beta$ is the birth rate and $\delta$ is the death rate. ...
1
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0answers
23 views

Particular solution of system of differential equations

Solve system of differential equations $$\begin{cases} x'(t)=2y(t)-x(t)+1 \\ y'(t)=3y(t)-2x(t). \end{cases} $$ My solution: First, I find the characteristic values $$\begin{vmatrix} 3-\lambda & ...
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0answers
3 views

the notion of Wazewski set in shooting method

I come across the notion of Wazewski set when studying shooting method for proving existence of a boundary value problem. Some authors who use shooting method to prove existence start with ...
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1answer
20 views

Tedious differential equation, solving for when $y(t) = 0$

If I have the differential equation: $$\begin{cases} \pi(2\,y\,R-y^2)\frac{dy}{dt}=-\pi\,r^2\sqrt{2\,g\,y} \\ y(0) = R \\ y'(0) < 0 \end{cases}$$ Where $r, R,$ and $g$ are all constants. If I ...
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1answer
12 views

Lie derivatives of vector fields and the binomial expansion

Given the Jacobi identity $[\mathbb{X},[\mathbb{Y},\mathbb{Z}]]+[\mathbb{Y},[\mathbb{Z},\mathbb{X}]]+[\mathbb{Z},[\mathbb{X},\mathbb{Y}]]=0$ and that the Lie derivative of a vector field is ...
0
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1answer
26 views

System of Non-Linear ODE

Does anyone have any clue of how to find an analytical solution for the following system: $$ \frac{dF_1}{dt}=(p+qF_1-rF_2)(1-F_1) $$ $$ \frac{dF_2}{dt}=vF_1(1-F_2) $$ $p$, $q$, $r$ and $v$ are ...
0
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2answers
46 views

Simple system of nonlinear ordinary differential equations

I'm trying to solve a system of ODEs of the form: $$\frac{d^2a}{dt^2} = \frac{-1}{(a-b)^2}$$ $$\frac{d^2b}{dt^2} = \frac{+1}{(a-b)^2}$$ and with the following boundary conditions: $$a'(0) = 0$$ ...