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
25 views

Curve fitting on non-linear ODE data

Background The graph below was generated by a set non-linear ODEs. For those of you who might want to know: It shows the maximum distance achieved by a cylinder when fired at a specified ...
2
votes
1answer
32 views

Solve the following symetrical differential equation

Recently I encountered a differential equation which is as follows: $\frac{d^3y}{dx^3} + x^3 \frac{d^2y}{dx^2} + 3x^2 \frac{dy}{dx} + 6xy + 6 = 0 $ I couldn't solve it because we were not taught how ...
1
vote
0answers
20 views

Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
2
votes
0answers
85 views

Periodic solution to DDE: $\frac{d}{dt}x(t)+\left(\frac{\pi}{2}+\epsilon\right)x(t-1)[1+x(t)]=0$

Consider differential equation with delay: $$\frac{d}{dt}x(t)+\left(\frac{\pi}{2}+\epsilon\right)x(t-1)[1+x(t)]=0.$$ Let's use $t=(1+c)\tau$ substitution to normalize time t: ...
1
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0answers
20 views

A specifc solution to Cauchy-Euler differential equation

Find $\alpha$ such as $y=x^\alpha$ is a solution to the differential equation $$x^2\frac{d^2y}{dx^2}+x(1-x)\frac{dy}{dx}-(x+1)y=0$$ (Oxford (2002), modified) We can derivate and obtaint ...
-4
votes
1answer
28 views

Solve the DE: $\dfrac{ dy}{dx} =\dfrac{y(x+y)}{x(y-x)}$ [on hold]

I worked it out and my answer came to be $y^3=x^3+3K$. Is my answer correct since I don't have any answer to this question. Thanks
2
votes
1answer
46 views

Numerically Solve a Second Order ODE with singular coefficients

I need to solve the following numerically: $$xy''+y'+xy=x$$ with initial conditions $y(0)=0$ and $y'(0)=1$. I need the solution for $x:[0, 10]$. I've written the ode as a system of first order odes ...
1
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0answers
26 views

Transform system to polar and sketch phase portrait. Show that $(0,0)$ is an unstable focus.

Transform the system $$x' = y - x(x^2+y^2-1)$$ $$y' = -x - y(x^2+y^2-1)$$ to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ...
2
votes
3answers
47 views

Numerical solution of an ODE system of equations using RK4

I have given an assignment to find the solution to the ODE system of equations as follow: $$\begin{cases} x_1' = x_1 + x_2 \\ x_2' = -3x_1 -10x_2 + x_2 ^2\end{cases}$$ With initial conditions: ...
0
votes
1answer
29 views

Value of $f'(3^{1/5})$ from the given differential equation

A function $y=f(x)$ satisfies $$xf'(x)-2f(x)=x^4 f(x)^2$$ and given that $f(1)=-6$ and $x$ belongs to all positive real numbers then prove that $f'(3^{1/5}) =8$ I have tried in this way...... Given ...
-1
votes
0answers
27 views

How to find the radius of convergence given recurrence relation?

Given, $$(1-x^2) \frac{d^2y}{dx^2}-x\frac{dy}{dx}+m^2y=0,$$ solving with power series, $$\sum_{n=0}^{\infty} c_nx^n,$$ then, the following recurrence relation is derived, ...
-2
votes
3answers
40 views

ordinary differential equation project suggestion [on hold]

My professor asked to write a project on ODE just to experience on how to write projects. It need not be a research project. Being in second rate school from third world country, we never did those ...
1
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2answers
27 views

Find the complete solution of differential equation

I´m having trouble with getting the complete solution for this equation. ${dy\over dx}=-y+2e^{-x}-1$ Somehow I´m not getting the same result as my answerlist or CAS tool. Thanks in advance. I´m ...
-1
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0answers
10 views

Exponential matrix using Laplace transform - reference request [on hold]

I am looking for a book that covers the concept of exponential matrix in detail using the Laplace transform plz
-1
votes
1answer
53 views

Need a hint using Runge-Kutta to solve this

Recall the fourth order Runge-Kutta method: $$x_{n+1} = x_n + \frac16(k_1 + 2k_2 + 2k_3 + k_4)$$ Apply it, with $h = 1$ to the initial value problem in the previous question to obtain a better ...
0
votes
1answer
36 views

eigenvaue of Sturm Liouville problem

Let the limit probem $$ \begin{cases} (P(x) y')' + q(x) y' + \lambda r(x) y=0\\ \alpha_0 y(0)+ \alpha_1 y'(0)\\ \beta_0 y(l) + \beta_1 y'(l) \end{cases} $$ with $\alpha_0^2 + \alpha_1^2 >0$ and ...
-3
votes
0answers
20 views

Determining the r value for a series solution [on hold]

I have trouble answering the 2nd part of the question. For the 1st part, I just simply plugged it into the derivatives and I end up getting $r_{1}=1/2$ and $r_{2}=1/3$. For the 2nd part, I tried ...
2
votes
1answer
75 views

Linear ordinary differential equations and their evolution operators for measurable operators

Consider the following homogeneous IVP: $$\begin{cases} \dot{u}(t)+A(t)u(t)=0 \\ u(0)=u_0 \end{cases}$$ for $u:[0,1]\to \mathbb{R}^n$ (some interval to some finite dimensional Hilbert space, let's ...
0
votes
1answer
26 views

How does $\text d/\text dx (\ln(1/x) + c) = \text d/\text dx (c -\ln(x))$?

I have to find the derivative of the following expression $\ln\left(\frac 1 x\right) + c$ it appears the answer is $-\frac 1 x$ however i am told it can be the original expression can be simplified ...
-1
votes
2answers
17 views

How can I convert this second order equation into a first order equation? [on hold]

In a previous exercise sheet, we were asked to transform the second order differential equation $$ x'' = -x + \alpha x^{3} $$ Into a first order equation. The solutions have since been released, but ...
-1
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0answers
17 views

Given recurrence relation, how to perform the ratio test? [duplicate]

Given, $$(1-x^2) \frac{d^2y}{dx^2}-x\frac{dy}{dx}+m^2y=0,$$ solving with power series, $$\sum_{n=0}^{\infty} c_nx^n,$$ then, the following recurrence relation is derived, ...
2
votes
1answer
149 views

Tough second order differential equation

I can't figure out this diff equation (in cylindrical coordinate). How can I solve it ? Any comments appreciated $$ \frac{1}{r}\frac{d}{dr}(r\frac{dE}{dr})+\frac{d^2E}{dz^2}+(\epsilon_0 ...
0
votes
2answers
23 views

Solving This Differential Equation

I need to solve this differential equation $${{{d^2}y} \over {d{x^2}}} - 2{{dy} \over {dx}} - 35y = - (x + 3)$$ I think I need to try a polynomial of the form $ax + b$ but I can't make progress past ...
4
votes
1answer
65 views

Which ODE Solution Method to Use?

It has been awhile since I've taken a course in differential equations, and I have problem, which requires I solve an ODE (after transforming a PDE) of the following structure: $$ f(x) - (x + ...
0
votes
1answer
25 views

Solving first order ODE

Anyone can help me to solve the two following ODE ? $\frac{dx}{dt} + \frac{1+\epsilon t - \epsilon x}{1+\epsilon t + \epsilon x}$ = 0 $\frac{dx}{dt} - \frac{1+\epsilon t + \epsilon x}{1+\epsilon t - ...
0
votes
1answer
17 views

Finding streamlines

Find the streamlines, particle paths and streaklines when $$u=xe^{2t-z}, \, \, \, v=ye^{2t-z}, \, \, \, w=ze^{2t-z}$$ What is the track of the particle passing through $(1,1,0)$ at time $t = 0$? To ...
1
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0answers
17 views

Solving a specific second order ODE

I need some one can help me to solve the following equation : $$z_{tt}-z_{xx}-2z_t = \alpha(t,x)(z_x-z_{tx})$$ where $\alpha(t,x) = \frac{4\epsilon(1+\epsilon t)}{(1+\epsilon t)^2 - \epsilon^2 x^2}$. ...
0
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2answers
46 views

PDE (linear, nonhomogeneous, first order)

Problem: $$\frac{du}{dx} + \frac{du}{dy} + u = e^{x+2y}$$ I have tried many methods and non have given me the correct result. The best lead I had was to change the coordinates and I got: ...
0
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0answers
25 views

Generating Function, finding a closed from solution and sequence [on hold]

I am having trouble solving the recurrence relation: $na_n = 3a_{n-1} - 4 a_{n-2} + \frac{8(3^{n-2})}{(n-2)!} $ So far I've used the generating function of A and gotten: $A'(n) = A(z)(3 + 4z) + ...
0
votes
2answers
19 views

Sketching solutions to IVP

Consider the following initial value problem (IVP): $$u_t + \cos(t)u_x = −u, \, \, \, \, (x,t) ∈\mathbb R×(0,∞)$$ $u(x,0) = u_0(x)$, $x ∈\mathbb R$, where $u_0 : \mathbb R → \mathbb R$ is a prescribed ...
0
votes
2answers
41 views

differentiate and solve $A = \frac{200}r + 3\pi r^2$

differentiate and solve $A = \frac{200}r + 3\pi r^2$ $A = 200r^{-1} + 3\pi r^2$ $A' = 6\pi r - 200r^{-2}$ $6\pi r - 200r^{-2} = 0$ From here I am not sure how to solve the equation with ...
1
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0answers
26 views

ODE: Solve $\frac{{\rm{d}}~}{{\rm{d}}x}\sqrt{f^2+e^{a x}+b}=f$ for constants, $a,b$ [on hold]

I would like to know the solution, $f(x)$, for the following ODE: $\frac{{\rm{d}}~}{{\rm{d}}x}\sqrt{f^2+e^{a x}+b}=f$ for some constant $a,b$.
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0answers
10 views

Solving a matrix of ODEs with an invariant of the matrix as a variable coefficient

I have the following system of ODEs: $$ \dot{\mathbf A} (t) + c \thinspace I(\mathbf A(t)) \thinspace \mathbf A(t) = \mathbf 0,$$ where $I$ is, say, the second invariant of the symmetric matrix ...
0
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0answers
26 views

How to use Newton's method for finding fixed points in Poincare maps.

As a homework I have to reproduce the numerical method given in the paper. Where there's the system $$ \dot{u}=f(u)+s(t)\\\\u=(u_1,u_2,u_3)\in\mathbb{R}^3$$ and ...
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0answers
6 views

How to deal with the composite function in a numerical approximation problem?

Consider a quasilinear two-point boundary value problem: $$-(a(u)u'(x))' = f(x) , x\in (0,1)$$ with $a(u)>0$ and $u(0) = 0, u(1) = 0$. I am supposed to derive an algebraic system so that I can ...
0
votes
1answer
32 views

Differential equation; Find the complete solution [on hold]

Hey guys I'm stuck with a Differential equation. Can anyone help me finding the complete solution to $dy/dx + 2y = 3e^x$ Any help would be greatly appreciated! :)
0
votes
2answers
19 views

Find fundamental set of solutions for 2nd order ODE?

I am asked to find the fundamental set of solutions ${y_1,y_2}$ for the equation $$y''-25ty'+25y=0$$ I am told that $y_1=t$ is a solution, how would I go about finding $y_2$, and if $y_1$ was not ...
0
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1answer
10 views

Differential equation: Find the equation for tangentline in P(1.3)

Hi guys Im in dire need of help with this one. A differential equation is given by (dy/dx)+(3x^2)*y=x^2 Define an equation for the tangentline for the graph at P(1.3) the particular solution goes ...
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0answers
20 views

Question about a solution for task involving differential equation

I've found a solution for one task. There's an equation $x^{`}=f(\frac{x}{t})$ and function $x(t) =ct $ is one of its solutions.And additionaly we know that $f^{`}(c)<1$ .The task is to prove that ...
0
votes
1answer
36 views

Newton's Law of Cooling

As shown in Figure 3.3.11, a small metal bar is placed inside container A, and container A then is placed within a much larger container B. As the metal bar cools, the ambient temperature $T_A(t)$ ...
2
votes
1answer
22 views

Solve a first order nonlinear ordinary differential equation

I have to solve the following problem: $y'=\frac{y \cos(x)}{(1+2y^2)}$ with the initial condition $y(0)=1$. I came up with the following equation: $y^2(x)+\log(y(x))=\sin(x)+c_1$. It is the first ...
6
votes
4answers
103 views

Power series solution for ODE

The ODE I have is $$y'(x)+e^{y(x)}+\frac{e^x-e^{-x}}{4}=0, \hspace{0.2cm} y(0)=0$$ I want to determine the first five terms (coefficients $a_0,\ldots, a_5$) of the power series solution ...
1
vote
1answer
19 views

What is the indicial equation of this differential equation?

The differential equation is $$x^2 y''+xy'+\left(x^2-\frac{1}{9}\right)y=0.$$ Using Forbenius' Theorem I am getting two indicial equations, which are: ((-1/9)r^(2))a0 =0, and ((-1/9)+2r+r^(2)+1)a1 ...
0
votes
0answers
13 views

What does the function $n(\gamma , z_{0})$ denote in this version of Cauchy Integral Formula?

In my lecture notes, the Cauchy Integral Formula for complex integrals is defined as $$ \int_{\gamma} \frac{f(z)}{z - z_{0}} dz = 2 \pi i \cdot n(\gamma , z_{0}) \cdot f(z_{0}) $$ What does the ...
0
votes
0answers
33 views

How to solve the differential equation $\frac{dy} {dx} =\sin(x+y)$ [closed]

I can't solve differential equation $\dfrac {dy}{dx} =\sin(x+y)$. How can I solve it?
3
votes
1answer
46 views
1
vote
1answer
43 views

How do I write $y''+y' +\sin y \cos y = 0$ as a first order system?

This would be fairly clear if $y = t$ where $t$ was the independent variable. But I can not see how this can be done.
2
votes
2answers
79 views

$f '' - (f ')^2 + f=0$; what is known about solutions?

I'm curious about solutions to the equation $$f''-(f')^2+f=0$$ on the whole real line, as well as solutions which are periodic. Any info about the obvious multivariable generalization would interest ...
0
votes
1answer
45 views

Differential Equation $\frac{d^2y}{dx^2}=\left( \frac{dy}{dx} \right)^2 + 1$

$$\dfrac{d^2y}{dx^2}=\left( \dfrac{dy}{dx} \right)^2 + 1 \quad \text{and} \quad y(0)=\dfrac{dy}{dx}(0)=0 \quad \text{on} \quad \left( \dfrac{\pi}{2},-\dfrac{\pi}{2}\right)$$ Let's define ...
1
vote
1answer
28 views

What trig identites were used in rewritting this equation

The undamped response for a system is: $$x(t)=x(0)e^{-\zeta \omega t}(\cos \omega_d t+ \frac{\zeta}{\sqrt{1-\zeta^2}} \sin \omega_d t)$$ In the book they claimed using trig identities they were able ...