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

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7 views

Particular solution of the differential equation “ y' + (2/3)y = 1-t, y(0)= y_0 ”

I have this particular differential equation: y'+(2/3)y = 1 - 1/2t y(0) = y_0 I have to find the specific value y_0 where the solution touches t axis, but it ...
-2
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0answers
18 views

Differential equations???

Two tanks of salt solution are connected to one another, with Tank 1 containing 30 gal of water and 25 g of salt and Tank 2 containing 20 gal of water and 15 oz of salt. Water with 1 g/gal of salt ...
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1answer
17 views

Use of undetermined coefficients issue

I'm given the problem $$y'' + 4y' = t$$ and asked to solve for y. I compute the general solution (using the characteristic equation) to be $$c_1 + c_2e^{-4t}\ ,$$ which I am pretty sure is correct. ...
0
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1answer
32 views

Solve the system of differential equations

I plan on adding more into later just a bit stuck, researching it at the moment. Solve the system of differential equations $$\begin{bmatrix} x'\\y' \end{bmatrix} - \begin{bmatrix} -11&15\\ ...
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2answers
26 views

Second order differential equations where rhs $= 6e^2\cos(3x)$

Solve the differrential equation $$y'' - 4y' + 13y' = 6e^{2x}\cos(3x)$$ where $y(0)=3$ and $y'(0)=-8$ I think we start like... For the homogenous case $$\lambda^2 -4\lambda + 13 = 0 $$ ...
1
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0answers
13 views

Finding particular solution to inhomogeneous system of differential equations

I am asked to find the general solution set of the following system of differential equations: $$\begin{cases} x' = 3x -2y-2 \\ y' = 6x-4y-1 \end{cases} $$ I found the general solution set of the ...
0
votes
4answers
37 views

First order differential equation: did i solve this equation right

So i'm trying to solve: $$x^2\frac{dy}{dx} + 2xy = y^3$$ I'm given this differential equation, that Bernoulli equation: $$\frac{dy}{dx} + p(x)y = q(x)y^{n} $$ I think i've solved it and ...
2
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0answers
23 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ and $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
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0answers
18 views

Newton backward and forward interpolation (for ODEs) intuition.

For Newton's backward and forward formulas, I understand everything algebraically, but can someone please explain me this formula intuitively, especially intuition how "powers of the forward ...
0
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0answers
8 views

Homogeneous and Nonhomogeneous ODEs - where the name comes from?

Why differential equations can be called Homogeneous and Nonhomogeneous? I understand equations behind these names, but where the word "homogeneous" comes from?
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0answers
29 views

upper bound of an $L^\infty$ function's derivative

Consider a function $u:\mathbb{R} \longrightarrow \mathbb{R}^n$ that is essentially bounded, i.e., $u \in L^\infty$. There is an upper bound of its derivative? I think there is not allways ( i.g. ...
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0answers
20 views

Trying to use the method “Stiff” (Rosenbrock method implementation) from the book “Numerical Recipes in C”.

The program is compilable but I don't think it works correctly. According to the book, we need also method "odeint" for adaptive stepsize adjustment and fully implement Rosenbrock method. I used the ...
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0answers
29 views

Solving Differential equation using Frobenius Method [on hold]

I want to solve a differential equation using the Frobenius method but unable to do it.Please anyone solve this for me.The equation is $$x(1+x)y''+3xy'+y=0$$
1
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1answer
36 views

First order differential equation: how do I prove that $u$ satisfies the differential equation

So I'm given this differential equation, that Bernoulli equation: $$\frac{dy}{dx} + p(x)y = q(x)y^{n} $$ now it says: Show that if $y$ is the solution of the above Bernoulli differential ...
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0answers
20 views

Applied mathematics for Clinical Medicine [on hold]

I'm a medical graduate, looking for advice/help on a project I would like to start. I would like to use applied mathematics to deconstruct the medical SOAP note into data sets that can be reproduced ...
0
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1answer
13 views

Need help with Laplace transform of piecewise /step functions

Hi I am having trouble figuring out how to calculate the laplace transform for $f(t)$ where $$f(t)= \begin{cases} e^{4t} & \text{if $ 0 \lt t \lt 2 $} \\ 1 & \text{if $ t \gt 2 $} ...
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0answers
30 views

Some results on Robin boundary conditions

I have the following boundary problem $$ (P): \left\{\begin{array}{l} y''(t) = p(t)\, y'(t) + q(t)\, y(t) + r(t),\\ y(t_1) = \alpha, \\ y'(t_2)+\gamma \cdot y(t_2) = \beta, \end{array}\right. $$ ...
-1
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0answers
25 views

Differential Equation ODE. [on hold]

Hello I have a problem with this differentials equations of first-order, im trying to do it with ode23 and ode23s. The differentials equations are the next one: y'+3y+z=0 z'-y+z=0 with this initial ...
1
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0answers
15 views

Why is this ODE solution only unique in either $(-\infty,0]$ or $[0,+\infty)$ and not in $\mathbb{R}$

Consider the following ODE: $$y'(t)=f(t,y)=e^{-t}+\log(1+y^2)$$ $$y(0)=0$$ You can clearly see the function is continuous on both variables, and the partial derivative with respect to $y$ is: ...
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2answers
48 views

Differential equation $y''-4y = e^{-x}$ [on hold]

I need help with the following differential equation: $$y''-4y = e^{-x}$$ (no initial conditions given) Any help is appreciated.
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0answers
15 views

Examples of ODEs with 3-dimensional function

I'm trying to test a numerical method program and I need some test cases, i.e. ordinary differential equations. I found some but in these examples the original Y funtion is unknown. I want to check if ...
5
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2answers
41 views

System of 3 differential equations

I'm trying to solve this system $$ \begin{align} x'&=x-3y+3z\\ y'&=-2x-6y+13z\\ z'&=-x-4y+8z \end{align} $$ must be reduced to a single equation I tried to express the x 3 and substitute ...
3
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1answer
36 views

problems with differential equation

i have problems solving eq. $$ u + \log(u-1) = \log (x); \quad u= \frac{y}{x}$$ which comes from solving diff equation $$x \frac{dy}{dx} - y= x\frac{y-x}{y+x}$$ any hints? thanks in advance
7
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4answers
280 views

solution to differential equation from deriving power series

Find the solution of the differential equation $$y'= 2xy$$ statisfying $y(0)=1$, by assuming that it can be written as a power series of the form $$ y(x)=\sum_{n=0}^\infty a_nx^n.$$ Im advised to ...
0
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2answers
15 views

A general solution of a partial differential equation with $f(x,y)$

I need to find a general solution to such a PDE: $$u_x-u_y=f(x,y)$$ I am able to find a solution if $f(x,y)=0$ or $f(x,y)=u$. But I have no idea how to get the general solution. Has anybody got any ...
1
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1answer
22 views

Differential equation where one solution induces a set of solutions

Consider a differential equation of the form: $$y' = f\left(\frac{y}{x}\right);\space\space\space x ≠ 0$$ where $f$ is any continuous function. I want to show that if $y(x)$ solves this equation, ...
0
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3answers
41 views

linearly independent (Linear algebra)

Show graphically that $y_1(x)=x^2$ and $y_2(x)=x|x|$ are linearly independent on $-\infty$ to $\infty$ but Wronskian vanishes at every point. The Wronskian is $$W = ...
2
votes
2answers
177 views

Laplace operator defined on a Sobolev space

Consider the Laplace operator $$A:W^{2,2}(\mathbb{R})\to L^2(\mathbb{R})\;\;\\A u = -u^{\prime \prime}$$ I want to know why this operator is closed (I'm using the closed graph theorem): Let ...
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0answers
33 views

Differential equation with steps [on hold]

What steps are involved in solving this differential equation? I found the answer at Wolfram, but it didn't show me how to get to the answer. $$y''+ y = \sqrt{x+y+1}$$
1
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1answer
21 views

How do we deduce that the initial value problem has always a unique solution?

Theorem - General solution of $y''+p(x)y'+q(x)y=0, x \in I (\star)$ Let $y_1, y_2$ be linearly independent solutions of $(\star)$ in an interval $I$. Then if $y$ is a solution of $(\star)$ in $I$, ...
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0answers
45 views

solve differential equation of $y''+y = \sqrt{x+y+1}$ [on hold]

I tried to solve this differential equation with no result, I even tried http://www.wolframalpha.com/ that showed no steps .. any help ?
0
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0answers
21 views

wave equation on a circular domain

Consider the wave equation for the displacement $$\text{u(r,$\theta $,t)}$$ in a circular domain $$\text{0 $<$ r $<$ a, -$\pi $ $<$ $\theta $ $<$ $\pi $}$$ How do I use the separation ...
3
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3answers
74 views

a linear differential equation with periodic coefficients

Let $$y' = a(x) y + b(x)$$ be a linear differential equation with continuous, periodic coefficients $a, b: \mathbb{R} \to \mathbb{R}$ that both have a period of $T > 0$. Also, we assume that ...
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0answers
25 views

What is the motivation for solving the Bessel equation.

My course is highly theoretical. For most part, we're taught to solve equations. But as a Physics student, I would very much like to know the motivation behind seeking the solution to the Bessel's ...
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2answers
24 views

Solving a two degree Differential equation (ordinary) with a variable coefficient

The question is to solve the following integral equation: $$y(x)=x-\int_{1}^{x}xy(t) dt; y \in C^1[1,\infty)$$ My try: I differentiated twice to get the ordinary equation $$y''(x)+xy'(x)+2y(x)=0$$ ...
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1answer
21 views

linear homogeneous constant coefficient systems [on hold]

Solve the following LHCC system by finding the eigenvalues, eigenvectors and generalised eigenvectors. Give a fundamental set of solutions and show that the set is independent. $$x'= \left[ ...
0
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1answer
16 views

Ordinary point of a Bessel DE

The Bessel DE: $$z^2\frac{\text d^2f}{\text{d}z^2}+z\frac{\text{d}f}{\text{d}z}+\left(z^2-m^2\right)f = 0.$$ The Bessel DE can be rewritten as: $$\frac{d^2f}{\text{dz}^2} + a(z)\frac{df}{ dz } + ...
0
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1answer
14 views

Reducing a Bessel's differential equation to a more 'useable' form

Suppose the given equation is: $$r^2\frac{\text d^2f}{\text{d}r^2}+r\frac{\text{d}f}{\text{d}r}+(\lambda r^2-m^2)f = 0$$ My text demonstrates the following: Let $$\text{z = }\sqrt{\lambda }r$$ So ...
1
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1answer
23 views

Question in regard to solving for inverse laplace transform

I am having some confusion when it comes to solving for the inverse laplace transform. ( We are allowed the tables with the common values by the way). Il give an example. Take, ...
0
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1answer
26 views

Differential equation for the logistic map

From the Wikipedia article on the logistic map I find the following definition as a recurrence relation: $$x_{n+1} = rx_n(1 - x_n) \tag{1} $$ Then, in another article, I see how to derive from this ...
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1answer
30 views

Change of variable of system of ODE [on hold]

I have one problem with the change of variables of this system: \begin{cases} 2y’ + z’ –y + 2z = 0 \\ y’ + 3z’ –3y +z = 0 \end{cases} with initial values $y(0) = 1$, $z(0) = 0$ I've made this ...
0
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0answers
17 views

System of ordinary differential equations, Fundamental Matrix

Let $\Phi(x,x_0)$ be a principal fundamental matrix of the system: $$u'=A(x)u$$ in an interval J. i.e. $$\frac{\partial \Phi(x,x_0)}{\partial x}=A(x)\Phi(x), \Phi(x_0)=I $$ Prove that: ...
2
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2answers
30 views

Proving a differential equation is a circle

So, I have solved the differential equation, to find the general solution of: $$\frac{y^2}{2} = 2x - \frac{x^2}{2} + c$$ I am told that is passes through the point $(4,2)$. Using this information, ...
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2answers
24 views

Differential equation maximal interval and solution [on hold]

Consider the differential equation $y' = 1 - y^2$. First, is $y(x) = 1$ the only constant solution? I now want to solve the equation for the initial value problem $y(0) = y_0$, with $y_0 > 1$. ...
2
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1answer
49 views

Differential equations application problem

I am studying differential equations, and I saw this interesting problem in another question (here): A destroyer is hunting a submarine in a dense fog. The fog lifts for a moment, discloses the ...
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0answers
21 views

Solving 2nd order ODE with 2 independent parameters(over finite intervals), with bounds on solution

I have a 2nd order ODE of the form: $\ddot {x} + 2c \dot {x} + 39Ex = 0 $ $Initial$ conditions being: x(0) = 0 and $\dot {x}(0)$=0.1 Where c is in the interval [1,5] and E is in the interval ...
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0answers
13 views

Characteristics and additional conditions for differential equation

I need to solve such a DE: $$(1+x^2)u_x+u_y=0$$ And then I need to draw its characteristics. The second part of the task says: Write three additional conditions such that this equation: Has one ...
2
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1answer
45 views

Explicit solution of parametric solutions of an ODE

I need to find the explicit solution of the following ODE: $y'+\sin y'=x$, $y=y(x)$. I have found these two parametric solutions: $x=t+\sin t$ and $y=\frac{t^2}{2}+t\sin t+\cos t+c$, $c\in\Bbb R$. ...
0
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1answer
27 views

What is the definition of ``2nd-order quasilinear parabolic'' ? for partial differential systems?

I have to know why the mean curvature flows are 2nd-order quasilinear parabolic. Let $\Omega\subset\mathbb{R}^n$ be a bonded domain (or a smooth manifold of $n$ dimensional) and $N\geq 2$. When the ...
2
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1answer
31 views

What is meant by a linear SDE?

I am sure this is a ridiculous question, but I can't seem to find a definition. I know the definition of linear ODE or PDE just by saying that the differential operator should be linear, but how does ...