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

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388 views

how to solve, $(x^2+y^2+2x)dx+2ydy=0$?

$(x^2+y^2+2x)dx+2ydy=0$ $f(x,y)=(x^2+2x)+y^2$ $g(y)=2y$ differentiating $f(x,y)$, w.r.t y, we have, $f_y^{'}(x,y)=2y$ differentiating $g(y)$, w.r.t x, we have, $g_x^{'}(y)=0$
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78 views

Ordinary differential equation Help!

Math-heads, I'm really struggling with the following ODE which has to be solved by the method of variation of constants and perhaps with some initial substitution: $\frac{dy}{dx} = ...
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59 views

Solve the differential equation $\frac{dy}{dx}=(x+y-1)^3-1 $

I am struggling with the following differential equation so any help would be much appreciated! Solve $\frac{dy}{dx}=(x+y-1)^3-1 $ by rewriting $z = x + y -1$ and solving for $z$ and then ...
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687 views

Existence and Uniqueness Theorem

I had a question about how to do one of these problems. So here's the question: Given this equation $y'=\frac{-\cos(t)y(t)}{(t+2)(t-1)}+t$, find if the initial conditions $y(0)=10, y(2)=-1, y(-10)=5$ ...
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52 views

Solution to ODE

I need to solve the following ODE $\mu p_tF^\prime(p_t)+\frac{1}{2}\sigma^2p_t^2F^{\prime\prime}(p_t)-rF(p_t)+Ap_t+b=0$ I know a general solution is on the form ...
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118 views

Solving a second order inhomogeneous differential equation with constant coeffcients

I a seeking to solve the equation: $$u'' + 2vu' + u = cos(\sigma t), \ u(0) = 1, \ u'(0) =0$$ where $0 < v < 1$. Then I have to show that the solutio is purely oscillatory (which I don't know ...
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23 views

expanding an equation , part of coordinate perturbation

if I have $$x_0 = \cos t $$ and I need to substitute $$ t = \tau + \epsilon T_1(\tau) + \epsilon^2 T_2(\tau) + \cdots $$ How do I go about the substitution and expansion to then gather like powers of ...
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903 views

differential equation : non-homogeneous solution, finding YP

hi i have a problem for this Differential Equations : $$ \frac{d^{3}y}{dx^3} - 9\frac{dy}{dx} = 10 - 4x $$ i know first we must solve the homogeneous equation: and my result is : $C_1 + C_2e^{3x} + ...
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58 views

$y'=(8\cos8x)/(3+2y)$ with $y(0)=-1$ Initial Value Problem: DiffEq

Im told to find the explicit form $y(x)$ from the given differential equation and its initial value. Then find where the solution $x=?$ attain a maximum. What I did was: $3+2ydy=8\cos 8xdx$ ...
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36 views

Differential Equation $y' = \frac{ty(4-y)}{(1+y)}$

$y' = \frac{ty(4-y)}{(1+y)}$ given $ y_0 = 2 $ At what time $t$ when the solution will first be 3.9 I tried solving this but it didnt work out so well. What I did was: separate dy/dx and move ...
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111 views

Set of differential equations modeling biological system

Im trying to describe a moleculatr biological system using some differntial equations, However, differntial equations is not my strong side. I'm thinking my equation set is quite trivial but i just ...
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61 views

Differential equation $uu''-u'^2=u'$

I have this: $$uu''-u'^2=u'$$ $$u'=\frac{du}{dx}$$ I can't solve it. I've done the recommended substitution $u'=v(u)$, so I have: $$u''=v'=dv/dx=\frac{dv}{du}\frac{du}{dx}=\frac{dv}{du}u'$$ ...
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315 views

Lipschitz Continuity

I have a quick question on how to interpret Lipschitz Continuity rather than normal continuity. What is the exact difference between the two, because I can't really find one. And is it more general ...
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127 views

Eulers Approximation Inductive Proof

Alright so I am having some difficulty with an inductive proof. I am attempting to prove the following: Given that the Euler Method is described by the given recursive formula: $y_{n} = ...
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1answer
96 views

Existence of Continuous Function

The following question requires the use of an easy theorem of calculus, but I am failing to see which one. Let $ g $ be non-constant and $ C^{1} $ on some interval $ I $. Show that for some ...
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240 views

An easy partial differential equation

I have just entered the study of ODEs. However, the professor, without having talked at all about it in class, asked us to solve the following partial differential equation: $\displaystyle ...
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117 views

Finding a differential equation from a known solution?

If $y = 6 + 3xe^x - \cos x$ is a particular solution of some homogenous differential equation, how we can find the corresponding differential equation? I know how to find the roots; for example ...
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411 views

Properties of ordinary differential equations - solution

I know that we take characteristic polynomial of forms like $\lambda^2 - \lambda - 2$ to find out the solution of ordinary differential equation of the form $e^{\lambda x}$ - conjugate ones can be ...
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74 views

Discrete Symmetry from an ODE

The next problem states that given a $m\times m$ non singular matrix $T$, such that $T^2=I$, is called a reflection for $\dot{x}=f(x) \Leftrightarrow f(Tx)=-Tf(x)$ for all $x\ \epsilon\ R^{m}$. Need ...
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64 views

Problem related to a differential equation

I am stuck with the following problem: Let $Y(x)=(y_{1}(x),y_{2}(x))$ and let $A$ is given by $$\begin{pmatrix} -3 &1 \\ k& -1 \end{pmatrix}.$$ Further, let $S$ be the set of values of $k$ ...
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378 views

Use Laplace transform to find a solution for $tx''+x'+tx = 0$

I was hoping I could get some help checking my work through. $\mathcal{L}\{x'(t)\} = sX(s)-1$ and $\mathcal{L}\{x''(t)\} = s^2X(s)-s$ Using the equality $\mathcal{L}\{-tf(t)\} = F'(s)$ we can ...
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245 views

Differential equations: $f(x,y) dx + g(x,y) dy = 0$

$$(2xy + 3y^2)dx + (x^2 + 6xy - 2y)dy = 0$$ $$y(1) = -1/2$$ How do you solve this? I have just started learning Differential equations and I have some trouble. Is this equivalent with this? ...
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443 views

How can you rewrite piecewise functions in terms of the unit step $u(t-a)$?

Consider $ u(t-a) = \begin{cases} 0, & \text{if }t<a \\ 1, & \text{if }t\geq a \end{cases} $ How can we rewrite a function like $ f(t) = \begin{cases} \cos2t, & \text{if }0\leq t \lt ...
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90 views

If the eigenvalues $\lambda_1$ and $\lambda_2$ for a linearized system are equal to $\pm bi$…

What does this suggest about the corresponding phase portrait at some critical point of the system? My textbook says "stable or unstable, center or spiral point," but how could I tell which it really ...
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2k views

Which of the following is gradient/Hamiltonian( Conservative) system

The question that I have to solve is found below. However, I do not know how to start the solution since I am unsure about the defintion of a Gradient/Hamiltonian System. What must I check first to ...
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94 views

(system of) nonlinear equations and instability

I heard that a system of nonlinear equations is unstable. I am curious of how "instability" is defined, and why do nonlinear equations show instability? Edit: OK, so what about contexts in matrices ...
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361 views

Repeated Root Eigenvalues

The question is: Solve the initial value problem: $$\begin{align*} \frac{dx_1}{dt}&=40x_1-6x_2+18x_3,\\ \frac{dx_2}{dt}&=-6x_1+45x_2+12x_3,\\ \frac{dx_3}{dt}&=18x_1+12x_2+13x_3,\\ ...
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964 views

Higher Order Linear Differential Equations - Solving for the particular solution

Given $y''' - 5y'' - y' + 5y = 3e^{-x}$, find the general solution. I found the roots for the homogeneous solution to be 5, 1, and -1: $$(r - 5)(r + 1)(r - 1)=0$$ $$y_h(x) = c_1e^x + c_2e^{-x} + ...
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603 views

Why is every solution to a homogenous second-order linear differential equation in the form $C_0e^{\alpha x} + C_1e^{\beta x}$

In textbooks, it's often casually mentioned, without explanation, that any two solutions added together is the general solution, the form of every other solution. I don't understand why this is or ...
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69 views

Basic differential equation proof

Show that if $u(t)$ solves $\dot{u} = Au$, then $v(t) = u(-t)$, solves $\dot{v} = Bv$, where $B = -A$. Similarly, show that if u(t) solves $\dot{u} = Au$, then $v(t) = u(2t)$ solves ...
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188 views

Asymptotic stability of fixed point

$f'(t)=af(t)(K-f(t))-bf(t)g(t)$ for $a,b,c,d,t,K>0$ $$g'(t)=cf(t)g(t)-dg(t)$$ This system has 3 fixed points (You can evaluate them if you set the 2 equations = 0). One point is ...
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225 views

Second order ODE with product of two functions on RHS

Find the general solution of the ODE $y′′ +16y=64x\cos4x.$ If $y(0)=1, y′ (0)=0,$ what is the particular solution? Attempt: I am just needing some help with the particular integral. I have tried ...
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112 views

Properties of a non-linear differential equation

I have some problems with the following differential equation, it looks a little bit confusing (because of notation), but please take a short look at it, it should be not too difficult. ...
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360 views

Higher order derivatives: variation of parameters

Higher order of derivatives: variation of parameters? $$y'''+y' = \tan t, \quad 0 < t< \pi.$$ Use variation of parameters to determine the general solution of the given differential equation. ...
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60 views

How can I show that $\det(v_1,v_2,\ldots,v_n)=dx_1\, dx_2\cdots dx_n(v_1,v_2,\ldots,v_n)$?

I wanted to use the definition of a wedge product which says $λ_1λ_2\cdots λ_k(v_1,v_2,\ldots,v_k)=\det(λ_i(v_j))$ with $1<i,j<k$ but I'm not sure if that even can work
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262 views

heat equation with polynomial coefficient

If I have $$u_t = a(x)u_{xx},\; x\geq 0$$ and initial value $$u(0,x)=u_0(x)$$ in the case of $a(x)=x^2$ with a change of variables $y=ln(x)$ it can be translated to the constant coefficient equivalent ...
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60 views

Finding the initial conditions which give the pendulum the greatest displacement

Given the motion of a pendulum modeled by: $x = c_1\left[ \begin{array}{cccc} \cos2t\\-2\sin2t\end{array} \right] + c_2\left[ \begin{array}{cccc} \sin2t\\2\cos2t \end{array} \right]$ What initial ...
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238 views

Write the general solution of the homogeneous problem for the system

We have in the form $x'' = Ax$: $\left[ \begin{array}{cccc} x_1''\\x_2'' \end{array} \right] = \left[ \begin{array}{cccc} -10&6\\6&-10 \end{array} \right]\left[ \begin{array}{cccc} x_1\\x_2 ...
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933 views

Finding general solution to system of ODEs using complex eigenvalues

Use the eigenvalue method to find the general solution to the initial value problem: $x_1' = 3x_1-5x_2$ $x_2' = 5x_1+3x_2$ $x_1(0) = 1$ and $x_2(0) = 4$ I found complex eigenvalues $\lambda=3-5i$ ...
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87 views

Can someone show me how to put together a 1st-order to represent population, which includes overcrowding?

I don't understand the sources I found online, so I'm hoping someone can show me how to do the following: I need to put together a 1st-order DE that models some population and includes birth rate, ...
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108 views

Use piccard iteration to find the solution of the ODE

Let $A\in M_{n\times n} (\mathbb{R})$. Use Picard iteration for $\dot x = Ax$, $x(0) = x_0$, to find the solution of this initial-value problem. I know that the solution must be $$ x = x_0 e^{At} = ...
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223 views

Solving $ y'' + (e^x - 1)y = 0 $

Find series expansion of the solutions to the following DE about $x = 0$. Try to sum in closed form any infinite series that appear: $$ y'' + (e^x - 1)y = 0 $$ My approach: OF course $x = 0$ is ...
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100 views

2nd Order Optimal Control Problem

I'm working on a homework problem in optimal controls and my plant model is described as: $$\ddot{x}(t) = u(t)$$ The performance index (cost function) is described by: $$J = 1/2\int_0^5u^{2}(t)dt\,$$ ...
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87 views

Any idea on how to solve this system of coupled ODEs?

I'm trying to find solutions for the system of ODEs $$ y_1'(t) = y_1(t)y_2(t) \\ y_2'(t) = 2y_2(t)^2 - y_1(t)^6 $$ And I'm assuming $ y_1(t), y_2(t) > 0 $. This comes from trying to find the ...
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877 views

Sixth order differential equation

Find the general solution of $y^{(6)}+2y^{(4)}+y'' = 0$. $r^6+2r^4+r^2=0$ $r^2(r^4+2r^2+1)=0$ $r^2[(r^2+1)(r^2+1)]$ So we have the roots: $0$: Multiplicity 2 $+i$: Multiplicity 2 $-i$: ...
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361 views

Finding the general solution of a sixth degree differential equation

Find a differential equation whose solutions are $y_1 = e^{2x} + e^{-4x}\sin(3x)$ and $y_2 = e^{-2x} + 5e^{2x}$. Am I supposed to assume that $y_1$ and $y_2$ can take the forms: $y_1 = Ae^{2x} + ...
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867 views

Find the equation of curve through $(1,1)$ the slope of whose tangent line at $(x,y)$ is $y^{10}/x^3$

... Express you answer as y^-9 = I assume I need to set up a differential equation, but I do not even know where to begin.
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177 views

To find closed form of $f(x)=\int_0^{\frac{\pi}{2}} e^{\sqrt{1-x^2 \sin^2 t}}\, dt$ as known functions

$$f(x)=\int_0^{\frac{\pi}{2}} e^{\sqrt{1-x^2 \sin^2 t}}\, dt$$ $u=\sin t$ $$f(x)=\int_0^{1} \cfrac{e^{\sqrt{1-x^2 u^2}}}{\sqrt{1-u^2}}\, du$$ $$f'(x)=\int_0^{1} \frac{-xu^2}{\sqrt{1-x^2 u^2 ...
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158 views

Show that $y(t) = t$ and $g(t) = t \ln(t)$ are linearly independent

I need to show that $y(t) = t$ and $g(t) = t \ln(t)$ are linearly independent. I thought I could use the Wronskian as follows: $y'(t) = 1$ $g'(t) = 1 + \ln(t)$ So $W(y, g) = (t)(1 + \ln(t)) - t ...
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2k views

Find two other linearly independent solutions to the second order differential equation

Fnd a general solution to the differential equation $y'' - y' - 2y = 0$. Then, use the two solutions you found to write two other linearly independent solutions to the problem. Write a second general ...