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

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1answer
36 views

Iterative trapezoidal method for differential equations

I am studying numerical methods for differential equations. I came accros the trapezoidal method in two forms, an explicit and an iterative one. I would like to know the advantages and disadvantages ...
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0answers
40 views

Numerical methods to solve Differential-Algebraic-Equations

I am new to the topic of differential-algebraic-equations: $ \dot x = f(x,u,c) $ $0=g(x,u,c) $ where $u$ are control variables and $c$ algebraic variables.In my first literature study i found two ...
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2answers
24 views

Find complex eigenvalues and eigenvectors then find the general real valued solutions

$A = \Bigg ( \begin{array}{cc} 1 & -2 \\ 2 & 3 \end{array} \Bigg )$ I have already found the eigenvectors and eigenvalues: $\lambda = 2 \pm i\sqrt{3}$ and $v_1 = [\frac{-1+i\sqrt{3}}{2}, 1 ...
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0answers
11 views

Drawing the characteristics from the method of characteristic

I have a question regarding the method of characteristics: I am given the following: $\frac{dS}{dt}+\frac{g(S)}{dx}=0$ $g(S)=0.5S^{2}$ $dS=\frac{dS}{dt}+\frac{dS}{dx} \frac{dx}{dt}$ From the ...
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3answers
45 views

Damped Harmonic Oscillator $2y''+8y'+8y=0$

Underdamped Harmonic Oscillator $2y''+8y'+8y=0$ I am given mass $m=2$, damping coefficient $b=8$ and spring constant $k=8$ I first need to change this into a first order system so I get: ...
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2answers
32 views

A hyperbolic solution to this differential equation

I have the following differential equation: $$\Big(\frac{dy}{dx}\Big)^2=\frac{y^2-A^2}{A^2}.$$ I am looking to obtain a solution $$y(x)=A\cosh\Big({\frac{x+B}{A}}\Big),$$ where B and A are constants. ...
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1answer
50 views

Generalized Hermite Function as eigenfunction of a differential operator

I'm going through this paper. The article defines function function $\phi_n^\mu(x)$ that is orthonormal on $L^2$ with measure $dm = dx$: \begin{equation} \phi^\mu_n ...
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0answers
6 views

How to implement these two boundary conditions in 1D differential discretization scheme?

$$\begin{cases} \left.\dfrac{\partial^3 h}{\partial x^3}\right|_{x=0} & =0\\[8pt] \left.\dfrac{\partial h}{\partial x}\right|_{x=0} &=0 \end{cases}$$ Nodes $x$ are from $0$ to $N$. The value ...
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0answers
21 views

Linear PDE with constant coefficients number of solutions

Lets say we have a linear PDE of n.order with constant coefficients. Is there a way to count the number of linearly independent solutions similar to an ordinary differential equation.
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1answer
32 views

Differential equation system IVP appears to be wrong $Y(0) = (2,5)$

Let $A = \Bigg ( \begin{array}{cc} 1 & -1 \\ 1 & 3 \end{array} \Bigg )$ Need to solve the IVP: $(x(0),y(0)) = (2,5)$ Solving for the eigenvalues get the characteristic polynomial: ...
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1answer
45 views

Solution to nonlinear ODE with square root

How do I solve the following equation? $\dot{x}=\sqrt{x^{2}-\frac{2}{3}x^{3}}$ with $x(0)=0$? I'm guessing I have to work with $dt=\frac{dx}{\sqrt{x^{2}-\frac{2}{3}x^{3}}}$ and integrate in [0,t'] ...
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1answer
51 views

Solve the integral $\int_0^1\left [\lambda_n\cos(\lambda_nx)+h\sin(\lambda_n(x)\right ]^2\mathrm{dx}$

Given $(\lambda^2-h^2)\sin\lambda=2h\lambda\cos\lambda \tag{1}$ Solve the integral $\int_0^1\left [\lambda_n\cos(\lambda_nx)+h\sin(\lambda_n(x)\right ]^2\mathrm{dx}\tag{2}$ The answer should be ...
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2answers
64 views

Induction Proof of Taylor Series Formula

I'm attempting to prove a formula for the taylor series of function from a differential equation. The equation is $$f(0)=1$$ $$f'(x) = 2xf(x)$$ I have found empirically that $$f(x) = ...
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0answers
33 views

(Differential Galois Theory) Where is the proof that the three-body-problem is unsolvable?

I'm looking for a proof, which shows that "the 3-body-problem" in physics is mathematically unsolvable. Does anyone know some URLs that contain a proof in mathematical detail? You know, in ...
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0answers
14 views

Determining the stability of zero solution [closed]

I do not know how to determine the stability of zero solution of the following differential equations $$\dot{x} = \sin (t), \qquad{} \dot{y} = -2 x - 3y$$
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2answers
46 views

Solutions of autonomous system $\dot{x} = f(x)$ if $f\circ T = -T\circ f$ for some nonsingular matrix $T$

Having an autonomous system $\dot{x} = f(x)$ with general solution $\phi(t, \xi)$. If $T$ is an $m \times m$ nonsingular matrix such that $f(Tx) = -Tf(x)$ for all $x\in \mathbb{R}^m$ prove $\phi(t, ...
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1answer
25 views

How to find periodic solutions for dynamical system?

I have the Hamiltonian system given by $$H=\frac{1}{2}x^2-\frac{1}{3}x^3+xy^2+\frac{1}{2}y^2$$ Using computer software I managed to plot the dynamical system in the phase plane. I am aware that the ...
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1answer
25 views

First-order nonlinear ODE similar to Bernoulli DE

I know that the Bernoulli equations, i.e. equations in the form $$ y' + p(x) y + q(x) y^{\alpha}=0$$ Can be easily solved with a change of variables. But what about equations in the form $$y' + ...
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1answer
66 views

Solve $\frac{dx}{dt}=\frac{at-\cos{x}}{at^2\tan{x}+t}$

Solve $\begin{align*}\frac{dx}{dt}=\frac{at-\cos{x}}{at^2\tan{x}+t}\end{align*}\\\\ $ Am I justified in doing the following substitution? If not, can a closed-form solution be found? Let ...
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1answer
34 views

How and why do we have to find $\frac{du}{dx}$ in order to solve the Euler-Cauchy equation?

Question: Use variation of parameters to find a particular solution to the inhomogeneous Euler-Cauchy equation, $$x^2\frac{d^2y(x)}{dx^2}-12x\frac{dy(x)}{dx}+42y(x)=-\frac{32}{x}$$ ...
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1answer
36 views

Solving differential equation using Laplace transform, problem finding inverse

Given $$y'' + 4y' + 5y = H(t-3)e^{-2t}, t>0, y(0) = 1, y'(0)=2 $$ To solve this diff. equation using Laplace transform. Seems very straightforward. On one side, we have $$\mathscr{L}\{y''+4y'+5y\} ...
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1answer
29 views

How to solve Inexact First-Order Ordinary Differential Equation

I have $xy' + y = \log x +1$.$~~~$ I started with $x\,dy + ( -\log x - 1 + y)\,dx = 0$.$~~~$ So $M(x,y) = x$ and $N(x,y) = -\log x - 1 +y$.$~~~$ Then I compute $\frac{\partial M}{\partial x} = 1$ and ...
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0answers
29 views

Linear algebra , magical square? [closed]

Let $T:\mathbb{R} ^{3}\rightarrow \mathbb{R} ^{3}$be a linear transformation . Prove the equivalance following statements : i) $\mathbb{R} ^{3}=ker\left ( T\right) \oplus im\left ( T\right)$ ii) ...
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1answer
53 views

Second-order linear differential equation with variable coefficients

Is there any way to find an analytic solution to the following differential equation? $$w'' - \left( \beta e^{-t} - 2 \right)w' + \gamma e^{-2t} w = 0$$ Here, $\gamma$ is negative and $\beta$ could ...
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0answers
11 views

Partizl differential Equations ?? [closed]

Partial Differential Equations 3. Consider the heat equation ∂u = ∂2u. ∂t ∂x2 (a) Show that if u(x,t) = tαφ(ξ) where ξ = x/√t and α is a constant, then φ(ξ) ...
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0answers
32 views

How to prove the orbit of planet is a circle or ellipse?

I think it is enough by $F=ma$ and $F=\frac{GMmr}{|r|^3}$.But I get stuck in a ODE $$ x'(t)=\frac{-GMx(t)}{(x^2(t)+y^2(t))^{3/2}} $$ How to deal it ? Or how to prove the orbit of planet is a circle ...
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1answer
21 views

Sine Curve Circular Transform - Parametric Equations

Is there a way to transform a sine curve so that the x-axis of the sine curve would become a circle, with the sine wave oscillating around the now-circular x-axis? What would be the parametric ...
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1answer
24 views

Rounding error of trapezoidal method

I'm working with the Modified Euler method sometimes called Heun's method or explicit trapezoidal method. I have a book on ordinary differential equations numerical analysis that claims: The ...
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2answers
33 views

Real Analysis equivalence for Differential Equations and Matrix Theory [closed]

I know that real analysis is the study for the proofs of calculus. But what's the equivalent for differential equations and matrix theory? I'd like to know for personal study. Can any good books be ...
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0answers
21 views

Given an ODE, how to find a specific value of the function? (without looking for the expression of the function if possible)

Let say we have an ordinary differential equation $$\frac{du}{dt}=k\,u(t)\,(1-u(t)),\quad t\geq0$$ that models the spread of a disease. where $u(t)$ is the proportion of animals infected by the ...
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0answers
28 views

stochastic differential equation exact solution

whats (is there) exact solution of (for) this sde? $dX_{t}=\mu X_{t}dt+\sqrt{\sigma X_{t}} dW_{t}$ and what's the distribution of that? thanks
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0answers
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A problem in differentiation. [closed]

Let $u(t,x)=f(t,x,{\dot x})-{\dot x}{\partial f \over \partial \dot x}(t,x,{\dot x})$ and $v(t,x)={\partial f \over \partial \dot x}(t,x,\dot x)$. Show that ${\partial v \over \partial t} = {\partial ...
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1answer
32 views

Frobenius Method recurrence relations

Q: By seeking a power series solution to $$2xy′′+(3−x)y′−y = 0$$ about $x=0$ show that there are two linearly independent solutions that have the recurrence relations $$a_{n+1} =\frac{a_n}{2n+3}$$ ...
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1answer
35 views

How to find the fundamental set of solutions of a second order ODE with constant coefficients, when given the solution form.

Q: By looking for solutions to $$y''' − y'' = 0$$ in the form $y = e^{rx}$, find a fundamental set of solutions to the above equation. A: {$e^0$, $xe^0$, $e^x$} ...
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2answers
37 views

Differentiating Query

Does the following make logical mathematical sense: $$x^2=t$$ $$\frac{d} {dy} (x^2)=\frac{d} {dy} (t)$$ $$2x\cdot\frac{dx}{dy}=\frac{dt} {dy} $$ $\mathbf{\therefore \frac{dy} {dx} =2x ...
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2answers
70 views

Dividing both sides by $y(x)$ when solving separable differential equations

Consider, for example, the differential equation $$\frac{dy(x)}{dx} = y(x)$$ This is generally solved as follows $$\frac{dy(x)}{dx} = y(x) \Longleftrightarrow \frac{1}{y(x)} \frac{dy(x)}{dx}= 1 ...
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1answer
33 views

Integrating both sides of an equation with respect to different variables [duplicate]

So im reading a book called "Ordinary Differential Equations" (Tenenbaum & Pollard) and in the introduction(ish) they are doing an example using a carbon dating problem, represented as: ...
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26 views

How to determine this system of ODE's?

I'm facing this problem: "Suppose you have this system of ODE's: $\begin{pmatrix} \dot y (t)\\ \dot x (t) \end{pmatrix} = \begin{pmatrix} a & b\\ c & d \end{pmatrix} \begin{pmatrix} y (t)\\ ...
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0answers
45 views

Solving integral inequalities ( Gronwall) [closed]

I don't know how to solve this inequality $$ v'(t) \leq ct +(v(t))^p, \qquad p >1 \, \quad \mathrm{and}\ \quad , c > 0 $$ With thanks.
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1answer
32 views

Does this function define first-order ordinary differential equations?

I've read 3 different books but, this condition seems to fit them all: It's homogeneous if: $y'=f(tx,ty)=f(x,y)$ Is this correct?
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3answers
55 views

Classical differential equation $mg-kv^3=m\frac{dv}{dt}$?

How to solve the differential equation $mg-kv^3=m\frac{dv}{dt}$? The equation is easily solvable when it is not $-kv^3$ but $-kv$ (linear) or $-kv^2$ (use trig identity). Strangely, I didn't find ...
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0answers
23 views

Solution to Cauchy-Riemann Differential Equation of Compact Support

I'm working through Forster's $\textit{Lectures on Riemann Surfaces}$ and am struggling with the following problem: Suppose $g \in \mathcal{E}(\mathbb{C})$ is of compact support. Prove there is a ...
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1answer
39 views

Differential equations question: Follow-up on Dynamical systems?

Yesterday I asked a question on here. Unfortunately I closed off the page without fully signing up for my account so I could not comment on the answer I received, whilst the answer was very good there ...
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0answers
18 views

Using Green's Functions in 3D to solve PDEs

I know that in 3D Green's functions are given by $G(x,\xi$)=$ \frac{1}{4\pi |x-\xi|}$. In my lectures we showed that in two dimensions, the solution for Poisson's equation with Dirichlet BCs is ...
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2answers
21 views

Differential equations with Euler's method

A differential equation y' + 2y = 2 - e^(-4*t) With starting point y(0) = 1 and increment ...
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0answers
22 views

Solve the following ODE using a Maclaurin expansion of the non-linear terms

Find two proper series solutions about the ordinary point $x=0$ of $$y''+e^xy'-y=0.$$ My proposed solution: Note that $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}.$ Assume there exists a power series ...
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0answers
34 views

Can you solve $y=\frac{a}{2}x^2\left(y'-\frac{1}{y'}\right)^2+x\left(y'-\frac{1}{y'}\right)+ax^2+c$?

I've recently come across this differential equation, but I am having trouble proceeding toward a solution. $y=\frac{a}{2}x^2\left(y'-\frac{1}{y'}\right)^2+x\left(y'-\frac{1}{y'}\right)+ax^2+c$ ...
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1answer
32 views

Analytical solution of a nonlinear ode

Is it possible to have some sort of analytical solution to this nonlinear ode (Corneal shape model) $${\frac {{\rm d}^{2}}{{\rm d}{\eta}^{2}}}f \left( \eta \right) -af \left( \eta \right) +{\frac ...
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1answer
36 views

Solve $-y''=e^x$ with $y(0)=y(1)=0$

Fine the displacement for an exponential force, $-y''=e^x$ with $y(0)=y(1)=0$ To solve this problem, I used integration twice gives me $$\int\left(\int -e^xdx\right)dx=-e^x+c_1x+c_2=y(x)$$ ...
0
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1answer
17 views

Writing up the solution for a nonhomogeneous differential equations system with complex Eigenvalues

The nonhomogeneous linear system of differential equations is given as: $$ x'(t)=Ax(t)+b $$ It has the following Eigenvalues and Eigenvectors: $$ \lambda=-1+i, \sqrt{2}, -1-i $$ $$ v_1 = ...