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

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43 views

$w''+w~ sin z=0$ solutions

I have $w''+w~\sin z=0$ and I want to sow that this equation has at least one solution of the form $e^{c z}v(z)$ where $v(z)$ is periodic My idea is to start by expressing $w_1(z+2\pi)$ and ...
1
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2answers
39 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 $y=e^{kx}$, $$k^2e^{kx}+9e^{kx}=0$$ ...
1
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0answers
16 views

second order linear ode in the complex domain

Consider $w''(z)+p(z)w'(z)+q(z)=0$ where $p(z), q(z)$ are analytic for $R\le|z|<\infty$ for some fixed $R$. Now I want to prove using analytic continuation of the solutions that the ode has one ...
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1answer
28 views
2
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0answers
31 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!
2
votes
2answers
29 views

An ODE with trigonometric coefficients

Anyone knows how to solve the following equation: $\cos(x) V(x) + \sin(x) V'(x) - V''(x) = 0$ with an arbitrary initial condition, let's say $V(0)= 1$. Thanks ;)
1
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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: ...
3
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2answers
112 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
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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
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1answer
15 views

How would you compute eigenvectors from this linear system?

I am stuck on a problem and I do not know how to obtain the eigenvectors: $\frac{dY}{dt}=\bigl(\begin{smallmatrix} -2&0\\ -3&1 \end{smallmatrix} \bigr)Y$ Work: I obtained the eigenvalues ...
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1answer
13 views

Find the function $f(x)$ when it satisfies the ode

The Fourier transform of the function $\frac{d}{dx}f(x)+xf(x)$ is $i[\frac{d}{dk}\widetilde{f}(k)+k\widetilde{f}(k)]$, so if a function $f(x)$ satisfies the ode $\frac{d}{dx}f(x)+xf(x)=0$, then the ...
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1answer
49 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.
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1answer
16 views

Method of Undetermined Coefficients: Solving for a particular solution

Solve for $2y''+3y'+y=t$ using method of undetermined coefficients. So I let $Y=At+b$ to solve for the particular solution. After substituting the first and second derivative into $2y''+3y'+y=t$, I ...
2
votes
1answer
36 views

solve the differential equation y'+sin(x)y=(sinx)^3, y(0)=-3 is the initial condition

I have a problem with the following differential equation, $y'+sin(x)y=(sin(x))^3$ First I have determined that the answer is something like: $y(x)=c(x)* e^{cos(x)}$, but now I have to determine ...
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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)
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0answers
41 views

How can i find the next approximate value with this iteration formula?

My ODE is : $$y'' + 2t(y')^2 = 0 $$ with initial values $$y(0)=2,y'(0)=1$$ and the analytical solution is $$y(t)=\tan^{-1}(t)+2 $$ which we convert to a system $$y_1' = y_2 \\ y_2'=-2t(y_2)^2$$ ...
3
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1answer
37 views

Analysis of stability of a linearized ODE with a periodic solution

I am asked to find the stability of the following ODE: \begin{equation*} \dot{y} = y^{2} + 2\cos(t)\sin(t) - \sin^{4}(t) \end{equation*} by linearizing around a particular solution $\eta = ...
2
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2answers
40 views

Ordinary differential equation $y'(t)=\sin(f(t,y))$

One whose solution never makes me happy is the following: $$y'(t)=\sin(y+t)\text{.}$$ I would start by substituting $z(t)=y(t)+t$ to get an ODE in $z(t)$, but then I'm not sure about how to substitute ...
2
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0answers
16 views

Lipschitz continuity in two variables [duplicate]

Prove that $y \mapsto f(x,y)$ is Lipschitz continuous, where $$f(x,y) = \frac{y}{x} \ln{\frac{y}{x}}, \ \ \ |x-1| \leq \frac{1}{2}, |y-1| \leq \frac{1}{2}e$$ I tried to solve this, but I find it ...
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1answer
27 views

How can I solve an ODE when $F(x_0)=F'(x_0)=0$ is given at an unknown point $x=x_0$ using bvp5c?

I'm attempting to solve the following ODE using MATLAB bvp5c. I've used bvp5c for other typical multipoint boundary value problems but I have no idea how to deal with ODEs with conditions given at an ...
0
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1answer
46 views

Solving a partial ODE

I have the following ODE: \begin{equation} \frac{dy}{dx} = 2u \end{equation} Above, $u$ is a function of $x$ and $y$. My gut tells me that this equation is separable. In this way, I have a ...
2
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2answers
51 views

Systems of Linear Differential Equations - Is this Correct?

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
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0answers
15 views

Is fourth order Runge-Kutta method validity

I wonder whether the fourth-order Runge-Kutta method is suitable for a second-order linear ODE with dissipative terms modelling free fall of an object through a viscous medium under the act of ...
2
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2answers
36 views

System of differential equations using substitution

Exact problem statement Solve the system $\left\{\begin{matrix} x_{1}'(t)=3x_{1}(t)-2x_{2}(t)+e^{2t},x_{1}(0)=a & \\ x_{2}'(t)=4x_{1}(t)-3x_{2}(t),x_{2}(0)=b & \end{matrix}\right.$ by using ...
3
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0answers
30 views

How to show no periodic orbits exist

I am trying to show that no periodic orbits exist for the system: $$ x_1'=y+x^2+xy^3$$ $$y'=-2x-y^3$$ I have tried using Dulac's criterion to find a function $g(x,y)$ such that $\Phi(x,y)$ given by ...
4
votes
3answers
139 views

A differential equation question

I totally have no idea about this... If $\frac{dy}{dx} = \sqrt{y^2+1}$, then $\frac{d^2y}{dx^2}$=? The correct answer is $y$.
0
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0answers
10 views

Determining the expressions of the coefficients of the full Fourier series from the Complex Series

Let $l \gt 0$ be a positive real number, and $\phi:[-l,l]\rightarrow\mathbf{R}$ be the function defined by: $$\forall x\epsilon[-l,l], \phi(x)=e^x $$ (1) Calculate the coefficients of the Full ...
4
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1answer
34 views

First eigenvalue of Laplacian and Poincaré inequality

Any idea on how to solve: $\int_{\Omega} |\nabla u|^2 d^n x=\lambda_1\int_{\Omega}u^2 d^n x$, with $u\in H^1_0(\Omega)$ and $\Omega\subset\mathbb{R}^n$, and $\lambda_1$ the first eigenvalue of the ...
2
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1answer
30 views

Use Green's function to find solutions for the boundary value problem

Find a solution using Green's functions $$y''+y=t; y(0)=0, y(1)=1$$ So far I have $$x(t)=c_1 \cos(t)+c_2 \sin(t)$$ so $$y_1=\cos(t), y_2=\sin(t)$$ and $$W(y_1,y_2)=-1$$ When I put that in the ...
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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 $\Delta X=2$ ...
0
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0answers
25 views

Stability of a linearized ODE with periodic solution

I'm asked to find the stability of the following ODE: \begin{equation*} \dot{y} = y^{2} + 2\cos(t)\sin(t) - \sin^{4}(t) \end{equation*} by linearizing around the particular periodic solution $\eta = ...
3
votes
2answers
77 views

Using the Jordan form Complex

Let $C$ be a complex $n \times n$ matrix with $\det C \neq 0$. Prove that there is a complex matrix $B$ such that $C = e^B$ Hint: use the Jordan form matrices for comlexas
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1answer
16 views

Differential Equations Show

A certain substance is formed in a chemical reaction. The mass of the substance formed t seconds after the start of the reaction is x grams. At any time the rate of formation of the substance is ...
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1answer
25 views

Showing that some differential equation has an infinite dimensional solution space?

I don't see how to proceed or even where to start to show this thing that I have found: The differential equation $$(\sin x)\frac{dy}{dx} - 2(\cos x)y = 0$$ has an infinite solution space of ...
0
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2answers
63 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
1
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1answer
29 views

Uniqueness of the solution to some differential equation.

I'm currently working on the subject mentioned in the title in a very general way. I think I get stuck for a stupid reason but here is my problem : I'd like to show that any solution to the equation ...
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1answer
26 views

Finding the series solution for a second order ode

Find the series solution for $y''-2y'+2y=0$ Assuming that $y=\Sigma^{\infty}_{n=0} c_nx^n$I got the recurrence relation: $c_nn(n-1)-2c_{n-1}(n-1)+2c_{n-2}=0$ Therefore: $c_3=\frac13c_1-\frac23c_0$ ...
0
votes
1answer
21 views

Differential equation by series solution method: equating coefficients to zero

I am following the solution for a problem, and I am stuck at the following equation: $$2a_2+\sum_{n=1}^\infty \left[(n+2)(n+1)a_{n+2}-a_{n-1}\right]x^n=0\tag1$$ Now, the professor equates the ...
0
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1answer
24 views

What is the significance of finding the series solution of a differential equation “about a point”?

I am learning the series solution method of solving differential equations, and I am curious as to what the rationale is for finding out the solution of the equation about a particular point. It seems ...
3
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1answer
41 views

Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
1
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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 ...
1
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1answer
34 views

Find equation of the curve

The product of the slope of the tangent line to a curve and sum of the coordinates of the point of contact equal to the ordinate for any point of the curve. This curve pass through the point $M_0 ...
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 ...
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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 ...
1
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1answer
28 views

Differential equation involving the Dirac delta

I have been trying to figure this out for a while, and I was wondering if anyone had any ideas. I need to solve the following differential equation: $m\frac{d^2 r}{dt^2}=\epsilon\delta'(r)$, where ...
0
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0answers
31 views

Find a solution to $x''+x=g(t)$, $x(t_0)=x_0$, $x'(t_0)=x_0'$

Find a solution to $x''+x=g(t)$, $x(t_0)=x_0$, $x'(t_0)=x_0'$, where $g$ is a continuous function in $\mathbb{R}$. Somebody can give me a hint?
1
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0answers
13 views

solving 1st order differential equation.

$$ \frac{\dot{c_t}}{c_t}=f'(k_t)-\delta-\sigma $$ $$ \dot k_t =f(k_t)-c_t-(n+\delta)k_t$$ $$(\delta,\,\sigma \text{ and } n \text{ are parameters}, \, c_t=c(t),\, k_t=k(t)) $$ How can I solve this ...
0
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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?
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0answers
13 views

Is there a program for convenient working with equations and coefficients?

I perform some calculations with one differential equation. Then I got a huge expression depending on $x$ and its degrees/powers. E.g. $$\alpha x+(4-x+\sqrt[3]{x})^2-(\beta\sqrt{x}+\frac12(x^3+1))^3 + ...