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

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0
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0answers
11 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 ...
6
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1answer
121 views
+50

Behaviour of solutions to ODE near singular points

I am having trouble understand how to classify what happens to solutions of ODE near singular points. For example; I have a question that is about the ODE given by; ...
-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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votes
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. ...
3
votes
1answer
52 views

Problems with a Simple Differential Equation

I am trying to solve the following: $y' = (y-5)(y+5)$ if $y(4) = 0$. So far, I have tried separating the variables and then use partial fractions and have followed these steps: (1) $A(y+5) + B (y-5) ...
0
votes
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\\ ...
1
vote
2answers
74 views

Sane solution for an ODE with physical interpretation

I have an object which is being subjected to a continual force that is a quadratic function of the object's velocity, ie, $F=f_0+f_1 v + f_2 v^2$ for arbitrary but given constants $f_0$, $f_1$, and ...
0
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0answers
35 views

Exact Similarity Solutions of System of Nonlinear Partial Differential Equations

I have been reading Self-Similarity and Beyond, by P. L. Sachdev. However, I am stuck on page 70, chapter 3, section 2. I have screen shotted the part which I am having a problem with I wonder if ...
5
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1answer
173 views
+50

Solution of $y''+xy=0$

The differential equation $y''+xy=0$ is given. Find the solution of the differential equation, using the power series method. That's what I have tried: We are looking for a solution of the form ...
5
votes
3answers
1k views

Converting Second Order Linear Equations to First Order Linear Equations

$\color{green}{\text{Question}}$: How can the following $\color{blue}{\text{second-order linear equation}}$ be converted into a $\color{blue}{\text{first-order linear equation}}$? This is our ...
1
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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 $$\lambda^2 -4\lambda + 13 = 0 $$ ...
3
votes
1answer
124 views

Why is subspace of solutions of linear ODE n dim?

If we are considering homogeneous linear ordinary differential equations among differentiable real-valued functions on $\mathbb{R}$, i.e. equations of the form $\mathrm{D} f =0$, then why is 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 ...
4
votes
2answers
62 views

How to address multiple cases in this BVP? (Laplace equation in quarter-annulus)

The original problem: $$\nabla^2 u =0 \ \ \ \ for \ \ \ 0<a<r<b\ \ \ ,\ \ \ 0<\theta <\frac \pi 2$$ $$u(r,0)=0,\ \ u(r,\frac \pi 2)=f(r),\ \ u(a,\theta)=u(b,\theta)=0$$ My ...
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
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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. ...
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 ...
0
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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
21 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 ...
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

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
vote
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 ...
5
votes
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 ...
0
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0answers
30 views

Some results on Robin boundary conditions

I have the following boundary problem $$ (P): \left\{\begin{array}{l} y''(t) = p(t)\, y'(t) + q(t)\, y(t) + r(t),\\ y(t_1) = \alpha, \\ y'(t_2)+\gamma \cdot y(t_2) = \beta, \end{array}\right. $$ ...
4
votes
1answer
338 views

Chebyshev Diff EQ

Find a power series solution about $x_0=0$ for the Chebyshev differential equation $$(1-x^2)y''-xy'+n^2 y=0,$$ as a function of of the integer $n$. Show that the solutions form a terminating ...
2
votes
1answer
33 views

The kernel $k(x,y)=\frac{y}{y^2+x^2}$ is a solution of which equation?

The kernel $$k(x,y)=\frac{y}{y^2+x^2}$$is a solution of (A) Heat equation (B) Wave equation (C) Laplace equation (D) Lagrange equation Which are correct ? I tried through ...
-1
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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
votes
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 $} ...
3
votes
2answers
543 views

General and particular solution of differential equation

1) I need to find, in implicit form, the general solution of the differential equation $$\frac{dy}{dx}=\frac{2y^4e^{2x}}{3(e^{2x}+7)^2}$$ 2) I then need to find the corresponding particular solution ...
-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 ...
3
votes
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 ...
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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votes
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.
-1
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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 ...
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, ...
3
votes
1answer
108 views

Differential Equations in Milnor's Topology from the Differential Viewpoint

On page $23$ Milnor states: Let $\varphi$ : $\mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function which satisfies $$\begin{cases} \varphi(x) > 0, & {\rm for}\,\|x\| < 1 \\ ...
3
votes
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
votes
3answers
75 views

The system of differential equations is in steady state

We have a system of non-homogeneous differential equations $$X'=AX+B$$ What does it mean that the system is in steady state?? $X$ is the vector $\begin{pmatrix} x_1(t) \\ x_2(t) \\ ...
2
votes
1answer
65 views
+100

Legendre Differential Equation, $y_1,y_2$ linearly independent solutions

$$(1-x^2)y''-2xy'+p(p+1)y=0, p \in \mathbb{R} \text{ constant } \\ -1 < x<1$$ At the interval $(-1,1)$ the above differential equation can be written equivalently $$y''+p(x)y'+q(x)y=0, ...
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 ...
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 = ...
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 $I$, ...
1
vote
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, ...
3
votes
2answers
59 views

Find two linearly independent solutions of the differential equation $(3x-1)^2 y''+(9x-3)y'-9y=0 \text{ for } x> \frac{1}{3}$

I want to find two linearly independent solutions of the differential equation $$(3x-1)^2 y''+(9x-3)y'-9y=0 \text{ for } x> \frac{1}{3}$$ Previously I have seen that the following holds for the ...
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 ...
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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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
vote
1answer
23 views

Light attenuation through water at an angle

I know that light intensity decreases exponentially governed by \begin{equation*} \frac{dy}{dx} = -ky \end{equation*} where $y$ is the intensity and $x$ is the distance. Now what happens when light ...
-3
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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 ?