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

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2
votes
1answer
69 views

Analytical Solution to Coupled Nonlinear ODEs

I am looking to solve several coupled nonlinear ODEs like this one: $\hspace{20mm} \frac{d x(t)}{dt} = C_1 \cdot x(t) + C_2 \cdot y(t) + C_3\cdot (x(t)^2 + y(t)^2) x(t),$ $\hspace{20mm} \frac{d y(t)...
-1
votes
0answers
57 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$

Is there any analytical solution for the following differential equation? $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$ A,B,C,D are non-zero constants and ...
1
vote
3answers
37 views

finding $k$ and $y(t)$

I am looking for help with this homework problem I am really stuck on. A function $y(t)$ is a solution of $$y′+ky=0.$$ Suppose that $y(0)=100$ and $y(2)=4$. Find $k$ and find $y(t)$. I worked it ...
0
votes
1answer
50 views

particular solution of the given differential equation

I need help with this calculus problem I am very confused about how to go through with this problem! Find the particular solution of the given differential equation $$\frac{\text{d}y}{\text{d}x}=−6xe^...
0
votes
0answers
24 views

Stability via Lyapunov Functions

Let $\sigma \gt 0$, $\tau \in \{-\infty\} \cup\mathbb R$ and $I = (\tau,+ \infty)$. Then define : $B_\sigma = \{x : |x|\lt \sigma\}$ $\bar B_\sigma = \{x : |x|\le \sigma\}$ $\color {blue} ...
0
votes
1answer
20 views

Could in-homogeneous ODE has more than one particular solution $x_p$?

I study ODE course at MIT open course, and the professor said several times "any particular solution would be fine". So, Could in-homogeneous ODE has more than one particular solution $x_p$ ? If so I ...
1
vote
0answers
56 views

In $x''+3x'+2x=e^{at}$, would $x_p$ at $a=-0.99$ looks like $x_p$ at $a=-1$? [closed]

In linear second order DE with constant coefficients: $$x''+3x'+2x=e^{at}$$ The characteristic equation of the DE is "$m^2+3m+2=0$" which roots are {$-1$,$-2$} The polynomial operator $p(D)=D^2+3D+2$ ...
1
vote
0answers
49 views

general solution of ODE, not exact

In http://math.jhu.edu/~szrebiec/images/exam1.pdf I found the exercises: 9) Solve the general solution to $(1+ty)e^{ty}+(1+t^2ye^{ty})\dfrac{dy}{dt}=1$ 10) Solve the general solution to $\left(\...
0
votes
0answers
26 views

How can isoclines show a solution to a differential equation?

Suppose we have this differential equation: $$y'= 2x + y$$ Now, the isoclines are: $$2x + y = m \implies y = m - 2x$$ How can I deduce the solution to this differential equation based on ...
0
votes
0answers
30 views

Complex fixed points on the bifurcation diagrams

I'm working with bifurcation diagrams, an extesion that is being made of them is the determination of complex fixed points in addition to the real fixed points, my question is: what information ...
0
votes
2answers
47 views

find the particular solution for the equation

I need help with this calculus problem. I dont know how to go about starting or finishing this problem Find the particular solution of the given differential equation $$\frac{dy}{dx}=−3\bigg(\frac{x^...
0
votes
4answers
73 views

solve $y$ as a function of $t$ in equation $y′=−8\frac{t}{y}$

I need help with this problem i have for homework i got an answer but it isn't right so i need help getting the right answer with some work Solve for y as a function of $t$ when $y′=−8\frac{t}{y}$ ...
0
votes
2answers
45 views

half life radioactive substance

I need help solving this calculus problem and i am really confused about how to work it Let $y(t)$ denote the mass of a radioactive substance at time $t$. Suppose this substance obeys the ...
0
votes
2answers
59 views

change in a coyote population

I am having a problem with this calculus problem: The rate of change of the number of coyotes $N(t)$ in a population is directly proportional to $650−N(t)$, where $t$ is the time in years. That ...
0
votes
1answer
23 views

a question about Initial value problem

I am trying to solve this Initial value problem below $$ x'_1(t)= 3x_1(t)+ x_2(t) - 7x_3(t) - 3x_4(t) \\ x'_2(t)= 3x_1(t)+ x_2(t) - 7x_3(t) - 3x_4(t) \\ x'_3(t)= 3x_1(t)+ x_2(t) - 7x_3(t) - 3x_4(t) \\...
0
votes
0answers
22 views

uniqueness of weak solutions for parabolic pde Evans

Hi I am trying to understand Evan's proof on uniqueness of weak solution in chapter 7. For the proof of theorem 4 below, I can see (35) and (36) make sense. But I have difficulty to see how Gronwall's ...
3
votes
2answers
65 views

Solution to the differential equation $xy''+y'+xy=0$

Show that the differential equation $$xy''+y'+xy=0$$ admits a solution of the form $$\varphi(x)=\int_0^1f(t)\cos(xt)dt$$ for some function $f(t)$. Since $$\varphi'(x)=\frac{d}{dx}\int_0^1f(...
-1
votes
0answers
33 views

How to solve $\frac{d\lambda}{dt} = \frac{2}{3T_v}(f(t)^2 - \frac{\lambda^3}{f(t)})$ where $f(t) = a+b \sin(wt)$

How can I solve $$\frac{d\lambda}{dt} = \frac{2}{3T_v}(f(t)^2 - \frac{\lambda^3}{f(t)})$$ where $f(t) = a+b \sin(wt)$ and $a, b ,w$ and $T_v$ are constants? I have to find $\lambda$ as a function of ...
0
votes
0answers
41 views

Incorrect answer - Simultaneous Differential Equations

The questions states solve for y such that $$y' = \begin{bmatrix} -4 & 2 & 1 \\ 1 & -3 & 1 \\ 3 & -3 & -2 \\ \end{bmatrix}y , y(0)= c = \begin{bmatrix} 1\\5\\3 \end{...
0
votes
0answers
18 views

Design with Matlab the equation of the position of a drone

I want to design with Matlab the equation of the position of a drone which is $u=m(\ddot z_{des}+K_pe+K_v\dot e+g)$ where $e$ and $\dot e$ can be calculated from the current and desired states $(z,...
0
votes
0answers
22 views

How to find the result of a differential equation in Matlab?

Given a variable $x$ defining the position of a drone, $\dot x$, its velocity, how do we find its acceleration $\ddot x$ in mathlab? I know that we can found the solution of a differential equation ...
-3
votes
1answer
61 views

How to solve $\frac{d^2y}{dx^2}+8\frac{dy}{dx}+16y=0$? [closed]

How to solve $\frac{d^2y}{dx^2}+8\frac{dy}{dx}+16y=0$ ? I found this question in a certain exam paper and the solution goes as follows. Auxillary equation is $m^2+8m+16=0$ $⇒(m+4)2=0⇒m=−4$ ∴ ...
0
votes
0answers
32 views

How to solve the differential equation $\frac{dy}{dx}-\frac{3x^2}{1+x^3}=\frac{\sin^2(x)}{{1+x}}$ and how to find its integrating factor?

How to solve the differential equation $$\frac{dy}{dx}-\frac{3x^2}{1+x^3}=\frac{\sin^2(x)}{{1+x}}$$ and how to find its integrating factor? The integrating factor is given as $$\frac{1}{1+x^3}$$....
1
vote
1answer
48 views

Stationary solution of a Fokker-Planck equation

I have a question that's driving me crazy. I have a Fokker-Planck equation $$\frac{\partial P}{\partial t}=x\frac{\partial P}{\partial x}+D\frac{\partial^2 P}{\partial x^2}$$ I look for the ...
0
votes
0answers
27 views

Bifurcations in two dimensional systems

Given the following autonomous differential equations, which illustrates an elliptic limit cycle, $r' = \alpha -\left(\frac{r}{a}\right)^2 \cos^2\theta - \left(\frac{r}{b}\right)^2 \sin^2\theta \\ \...
0
votes
0answers
16 views

Implicit Euler using Taylor

I was reading script about differencial equatations. More specific about schemes that help calculate them - implicit Euler. That method was analyzed using something similar to Taylor but i am not sure ...
1
vote
2answers
39 views

Differential equation with one derivative: $y'=y\cos(x)+x\cos(x)-1$

$y'=y\cos(x)+x\cos(x)-1$, I tried to make it in the form $ay''+by'+c=0$, but I can't find the roots.
1
vote
1answer
53 views

Differential equation, Solution is a Bessel fucntion

this is my first post here. I knocked my head on a differential equation yesterday, this one: $$ \frac{12 \nu}{x^2} \frac{S(x)''}{S(x)} = -\lambda^2 $$ Where $nu$ is a constant. The book says the ...
4
votes
1answer
42 views

First order differential equation solution [closed]

I couldn't solve the following problem, can you please help? $$y' =\tan (x+y) -1$$ $$(x^4 -2xy^2 +y^4 ) dx - (2x^2 y -4xy^3 )dy = 0$$ $$(y \sec^2 x +\sec x \tan x ) dx +(\tan x +2y )dy =0$$
2
votes
2answers
60 views

Differential equation questions

I am studying differential equations of order $1$ and $2$ and I had these questions on my mind: 1. Is it true that every differential equation has infinitely many solutions if there are no initial ...
1
vote
1answer
43 views

finding the general solution of linear system

Under what conditions does the solution of the following system exist? How can one find the general solution of the linear system $$\frac{dX(t)}{dt}=A(t)X(t)+B(t)$$ where: - $A(t)$ is an $n \times n$...
-1
votes
0answers
35 views

Differential Equations confused over question

Interpret the statement as a differential equation. On the graph of y = \Phi(x) , the slope of the tangent line at a point P(x, y) is the square of the distance from P(x, y) to the origin. From my ...
1
vote
2answers
27 views

Trajectories of Differential Equation Systems with Complex Eigenvalues

In an autonomous system of 1st order differential equations in order to find the trajectories one must solve $$\frac{dy}{dx}=\frac{Ax+By}{Cx+Dy}$$ In the case of complex eigenvalues my notes say ...
2
votes
1answer
85 views

Proving $J_n(x)N_{n+1}(x)-J_{n+1}(x)N_n(x)=-\dfrac{2}{\pi x}$: Part $2$ of $3$

The following question is the second part to this previous question: Prove that $$J_p(x)J_{-p}^{\prime}(x)-J_{-p}(x)J_{p}^{\prime}(x)=-\frac{2}{\pi x}\sin(p\pi)\tag{1}$$ from $$\frac{\mathrm{d}}{\...
2
votes
1answer
50 views

Three coupled differential equations

I am trying to solve the following set of differential equations, I am trying to do it by the usual decoupling methods like adding the equations, subtracting etc which makes the process rather lengthy....
1
vote
0answers
16 views

Integral and differential inequality

I have integral and differential inequality $y'(t)<Ch^{k+1}+\int_0^ty(s)ds+y(t)$ where $C,h$ are constants and $y$ is positive function with y(0)=0 My goal is to prove $y(t_F)<Ch^{k+1}$ ...
0
votes
0answers
24 views

Formal approximation for second-order ODE with varying coefficients

I have a differential equation of the form $$0=a+by(x)+cf(x)+z(x)f''(x)$$where the functions $y$ and $z$ are known and we want to find $f$. If $z$ is constant, i.e. $z(x)=Z$, it is straightforward to ...
0
votes
1answer
23 views

Proof that linear difference operator, $(σ-1)^{k+1} (p) = 0$ for all $p$ $\epsilon$ $\mathbb{Q}[t]$, with $deg(p) \leq k$.

I am trying to prove that linear difference operator, $(σ-1)^{k+1} (p) = 0$ for all $p$ $\epsilon$ $\mathbb{Q}[t]$, with $deg(p) \leq k$. In this case $\sigma(t)=t+1$ and $\sigma($anything else$)=$...
0
votes
0answers
26 views

A proof of the test of exactness for differential equations

I went through a proof of the following theorem for test of exactness of differential equations: Let the functions $M(x,y)$, $N(x,y)$, $M_y(x,y)$, and $N_x(x,y)$, be continuous on the region $R=\{(x,...
1
vote
1answer
32 views

Integrating factors, a missing solution

I want to solve the differential equation $(3xy+y^2)+(x^2+xy)y'=0$. If I use the integrating factor $\mu (x)=x$ so that the original differential equation becomes exact, then the general solution that ...
1
vote
0answers
25 views

Solving n-dimensional first order linear pde

While working on a problem in game theory, I'm stuck at a problem which requires me to solve the following linear first order PDE on $K$ independent variable: $\sum_{k=1}^K(\frac{\partial u}{\partial ...
0
votes
1answer
77 views

Examining a solution of a differential equation without knowing the solution

The differential equation is given by $$\dot x=-x \cos x$$ with $x(0)=x_0\in(0,\frac{\pi}{2})$. Now I need to show that for each choice of $x_0$ the domain of the solution $x: I\rightarrow \mathbb{...
2
votes
1answer
81 views

Solve differential equation by using polar coordinates

For $\alpha, \beta>0$ the differential equation, I am trying to solve, is given by $$\begin{pmatrix}\dot x_1\\\dot x_2\end{pmatrix}=\alpha\sin(x_1^2+x_2^2)\begin{pmatrix}x_2\\-x_1\end{pmatrix}+\...
0
votes
3answers
68 views

3rd order differential equation with variable coefficients

How to do I solve this differential equation? $$ x^3 u′′′ + x^2 u′′ + x u′= 0. $$ The series solution method is not working in this case.
0
votes
1answer
25 views

How to solve a nonlinear second order differential equation?

I have been trying to find ways to solve: $$J\frac{d²\theta(t)}{dt²}-K_m cos(\theta(t))=-\tau_f$$ With the initial conditions $$\theta(t=0)=0$$ $$\frac{d\theta}{dt}(t=0)=0$$ Without success. Is that ...
2
votes
4answers
77 views

Find an equation of the curve that passes through the point $(0, 6)$ and whose slope at $(x, y)$ is $\frac{x}{y}$. Book wasn't helpful.

I am using James Stewarts Early Transcendentals Calculus, and Section 9.3 (which is where this problem comes from) doesn't seem to have anything remotely similar to the problem I am facing. No ...
2
votes
1answer
78 views

Proving $J_n(x)N_{n+1}(x)-J_{n+1}(x)N_n(x)=-\dfrac{2}{\pi x}$: Part $1$ of $3$

This is the first part of a proof that $J_n(x)N_{n+1}(x)-J_{n+1}(x)N_n(x)=-\dfrac{2}{\pi x}$: Write Bessel's equation $$x^2y^{\prime\prime}+xy^{\prime} + (x^2 - p^2)y=0\tag{1}$$ with $y=J_p$ and ...
0
votes
1answer
29 views

Solving $\sin(\sqrt{\lambda}L) + \beta \cos(\sqrt{\lambda}L)\sqrt{\lambda} = 0$

I'm working with the ODE $$-\frac{d^2u}{dx^2}=\lambda u$$ and trying to find eigenvalues and eigenfunctions corresponding the boundary conditions $$u(0)=0, u(L)+\beta \frac{du}{dx}(L)=0$$ Assuming ...
0
votes
1answer
37 views

Differential Equation using Laplace transformation.

I have a problem solving this differential equation using Laplace transformation. $y'' -9y=0 , \ y(0)=1 , \ y'(0)=0$