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

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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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2answers
37 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 ...
-3
votes
1answer
52 views

How to find a general solution using substitution for $\frac{dy}{dx}$ + $e^{2x−3y}$ = −$e^x$ [on hold]

Using the substitution $y=\frac13 \log f$ to find general solution for $\frac{dy}{dx}$ + $e^{2x−3y}$ = −$e^x$
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0answers
22 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
5 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
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1answer
16 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 ...
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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
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! :)
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0answers
15 views

Solving constrained Euler-Lagrange equations with Lagrange Multipliers (Geodesics)

I'm trying to solve a calculus of variations geodesics problem using Lagrange Multipliers, showing that the geodesics of a sphere are the so-called great circles. I am using a constrained Lagrangian ...
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 ...
2
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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 ...
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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 ...
1
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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 ...
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0answers
32 views

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

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

How to find a general solution for $\frac{dy}{dx}$ + $e^{2x−3y}$ = −$e^x$

$\frac{dy}{dx}$ + $e^{2x−3y}$ = −$e^x$ Do I use integrating factors?
1
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1answer
12 views

Writing a sum of unit step functions as a piecewise function

After taking the inverse Laplace transform of the following $$\mathcal{L}^{-1}\{G(s)\}=\mathcal{L}^{-1}\left\{\frac{e^{-2s}+e^{-3s}}{s^2-3s+2}\right\}$$ I have ...
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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 ...
1
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1answer
13 views

Compute Heaviside Laplace transform, then use this to solve initial value problem

I've been stuck on this problem for a while, and can't really seem to find where I should go with it, or where I went wrong if I made a mistake. Let $L(x)$ denote the Laplace transform of x. Q: ...
1
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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 ...
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0answers
17 views

Second Order Linear Non-Homogeneous DE solution with Power Series $x^2y'' - 4xy' + 6y = x^2 \cos x$

My instructor wants me to solve the above equation using power series and another method, and then to confirm the results are the same This equation does not have constant coefficients and a can't ...
0
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1answer
20 views

Dirac Delta Function Problem

In my Differential Equations class, we had the following equation on a test today: $y''+6y'=2\delta(t)$, $y(0)=0$, $y'(0)=1$. I got the following using Laplace Transforms (the only way I know how to ...
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0answers
14 views

A quick question on graphing Heaviside functions

I'm trying to graph the function $f(t) = \sin(2(t-\pi))\mathcal{U}(t-\pi)$, where $\mathcal{U}$ is the Heaviside function defined by: $\mathcal{U}(t-a) = \begin{cases} 0 &\mbox{} t \lt a \\ 1 ...
0
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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 ...
0
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1answer
22 views

Differential equations describing a physical system - deriving the equations and elaborating on such equations

We see the mass-and-spring system above and we derive two systems of differential equations to describe its behavior. For example, we have $$m_2 y^{\prime \prime} = k_2 x - (k_2 + k_3)y$$ I have ...
1
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1answer
42 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 ...
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0answers
13 views

RK4 for 2nd order ODE: multiplication by $h$

Take a first-order ODE $y' = f(x,y)$ for instance, with $y_0 = y(x_0)$. To calculate $y_1$ with RK4 one must first calculate $k_1 = f(x_0,y_0)$, and so on. Now look at the accepted answer (and ...
2
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1answer
35 views

Solving a differential equation: $y^4 \left(\frac{dy}{dx}\right)^4 = (y^2-1)^2$

How would one go ahead solving the following for these conditions: (i) passes through $(0,\frac{\sqrt{3}}{2})$ and (ii) $(0,\frac{\sqrt{5}}{2})$ : $$y^4 \left(\frac{dy}{dx}\right)^4 = (y^2-1)^2$$ ...
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0answers
35 views

Solve $(x^2+2x)\frac{d^2y}{dx^2} -2(x+1)\frac{dy}{dx} +2y=0$ by series and elementary method [on hold]

Solve the following differential equation by series and elementary method: $$(x^2+2x)\frac{d^2y}{dx^2} -2(x+1)\frac{dy}{dx} +2y=0$$
0
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1answer
46 views

Differential equation $\frac{dy}{dx}=f(y),y(0)=y(1)=0$ [on hold]

Differential equation $$\frac{dy}{dx}=f(y),y(0)=y(1)=0$$ where $f:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous. Then $1.$ $y(x)=0$ if and only if $x\in (0,1)$ $2.$ $y$ is bounded $3.$ ...
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0answers
11 views

Weak formulation of non-local Neumann problem

Consider the following probleblem: $$ -\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in $\Omega$} $$ $$ \partial_{\nu}u=0 \qquad \text{in $\partial\Omega $} $$ where $$\Omega\quad ...
1
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1answer
17 views

Mean Value Property - Nonnegative Harmonic function

I want to prove that the mean value property $$u(\textbf{x}_0) = \frac{1}{\pi r^2} \int \int _{\left \{ \left | x_0-x \right < r| \right \}} u(\textbf{x})d\textbf{x}$$ for non-negative harmonic ...
2
votes
2answers
78 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 ...
1
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2answers
37 views

$25 w(w-1)y''+(14-15w)y'+y=0$ - Gauss's Hypergeometric equation

I would like to solve the equation $(x^2-x-6)y''+(5+3x)y'+y=0$ near the singular point $x=3$. I think we have to solve this problem in considering the Gauss's hypergeometric equation on the form ...
1
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1answer
30 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 ...
0
votes
1answer
18 views

Free oscillations and natural frequencies question

I'm given a mass-and-spring system with a couple second-order differential equations describing the behavior of the system. Those are: $$x^{\prime \prime} +50x - \frac{25}{2}y = 0$$ and $$y^{\prime ...
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0answers
43 views

Differential equations - Hypergeometric function [duplicate]

I would like to solve the equation $(x^2-x-6)y''+(5+3x)y'+y=0$ at $x=3$. I think we have to solve this problem in considering the Gauss's hypergeometric equation on the form (*) ...
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1answer
19 views

Motion of a pendulum equation in the George Simmons book on differential equations [on hold]

I just can't understand the transition between this two formulas, why $dt$ becomes $T/4$. Can anybody help me with that?enter image description here
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1answer
18 views

Can this differential equation be transformed into an hypergeometric equation?

$$(1+x^2)y'' -4xy' + 6y = 0 $$ Can this be transformed into an hypergeometric equation of the form $x(1-x)y'' + (c - (a + b + 1)x)y' -aby = 0$? I know that we can do the transform is the term before ...
0
votes
1answer
8 views

Real solutions of $C_1e^{jx}+C_2e^{\bar jx}, C_1,C_2\in \mathbb{C}, j=\frac{1}{2}+i\frac{\sqrt 3}{2}$

Why does $C_1e^{jx}+C_2e^{\bar jx}, C_1,C_2\in \mathbb{C}, j=\frac{1}{2}+i\frac{\sqrt 3}{2}$ has the following result: $$S_{H_2}(\mathbb{R})=C_1e^{-1/2x}\cos(\frac{\sqrt ...
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0answers
53 views

How to solve this differential equation

I am unable to solve this equation. Please give me the way, how can I solve this? $$ y'''\sin x - y'e^x\cos x = 2 $$
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1answer
25 views

How to find Integrating factor of this equation

Equation $$ (x^2 e^{-y/x} + y^2 ) dx -xy dy =0 $$ Not Exact Q) Further solution is possible ? means Integrating factor
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0answers
28 views

Heat problem with an internal source of heat for which the maximum principle doesn't hold.

Heat problem with an internal source of heat for which the maximum principle doesn't hold. The problem is the following and honestly I don't know how to solve it... $$u_{t}=u_{tt}+2(t+1)+x(1-x) , ...
1
vote
1answer
36 views

How to graphically represent $\ddot x$?

We know that given a differential equation: $$\dot x = f(x), x \in X$$ The $\dot x$ is understood as the tangent vector on the solution trajectory $x$ lying in the tangent space of $X$ What about ...
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0answers
34 views

Find the solution of differential equation

This is a differential equation. I tried it many times but cant solve . xy'+(1+x)y=e^{-x}*sin2x My solution After finding ...
0
votes
2answers
41 views

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series. I know that $e^{x^2}=\sum_{n \geq0} \frac{x^{2n}}{n!}$, but I don't know how to find the coefficients $a_n$ in considering ...
0
votes
0answers
21 views

Spatial derivative of pendulum to time derivative

$m\ddot{x}\dot{x}+\frac{dV}{dx}\dot{x}=0\Rightarrow \frac{d}{dt}\left [ \frac{1}{2}m\dot{x}^{2}+V\left ( x \right ) \right ]=0$ $Using \frac{dV\left ( x\left ( t \right ) \right ...
2
votes
1answer
38 views

How to solve a differential equation

How would you solve this differential equation? I have been trying to use the technique of separable equations, but haven't got very far. $$\frac{D'(x)}{D(x)}=\frac{x}{1-x}$$
0
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2answers
10 views

Show that if $c$ is a positive constant, then $\frac{1}{c}F(\frac{s}{c})=\mathcal{L}\{f(ct)\}$

Suppose that $F(s)=\mathcal{L}\{f(t)\}$ (Defined as the usual Laplacian operator). Show that if $c$ is a positive constant, then $\frac{1}{c}F(\frac{s}{c})=\mathcal{L}\{f(ct)\}$. ...