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

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5
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1answer
39 views

Tough Differential Equation

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
1
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0answers
18 views

a question regarding wronskian

I was working on following problem: Let $y_1$ and $y_2$ be solutions of $x^2y'' + y' + (\sin x)y = 0$ satisfying $y_1(0) = 0, y_1'(0)=1,y_2(0) = 1, y_2'(0)=0 $. I worked like following: since ...
1
vote
0answers
17 views

Limit to infinity from a differential equation

Let $R'(t) + \nu R(t) = \nu F(t)$, $F(0)=0$, $R(0)=0$, $f(t) \geq 0$, $F(t) = \int_0^t f(\tau)d\tau$, $F(t) \leq 1$, and $\lim_{t \rightarrow \infty} F(t) = 1$. I solved the differential equation and ...
1
vote
1answer
25 views

McLaurin series expansion to evaluate a function

I have a maths assignment due for college based on the McLaurin series and don't understand how to do it. I need to use a McLaurin series expansion to evaluate a function. The function is the ...
1
vote
1answer
26 views

How do we know radioactive decay can be modeled by the half-life equation, dq/dt = -aq?

I understand how to solve it. but why does $$\frac{d \lambda}{dt} = -k \lambda$$ The equation, in and of itself, means the rate of decay is proportional to the amount at a given time. How do we know ...
2
votes
0answers
19 views

Differential equation: $A(x)y''(x)+A'(x)y'(x)+y(x)/A(x)=0$

So give the differential equation $$A(x)y''(x)+A'(x)y'(x)+\frac{y(x)}{A(x)}=0,$$ with $A(x)$ a known function and $y(x)$ te be determined. What is the solution for this differential equation ? I've ...
0
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0answers
15 views

Existence & uniqueness of a second order ODE

Details on the model can be found here under III titled "Will The Valve Hold?". ...
2
votes
1answer
26 views

example which doest not satify Lipchitz condition but has unique solution

$y'=1+\sqrt y , y(0)=0 $ Show that this IVP does not satify Lipchitz condition but has a unique solution. I have shown the first way, like this: Let $f(x,y)=1+\sqrt y $.Then $\frac ...
0
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0answers
27 views

Convert the following system to a first order system:

Really having a hard time with this.....Convert the following system to a first order system: $$\frac{d^2x}{dt^2} -3\frac{dy}{dt}+x=\sin(t)\\ \frac{d^2y}{dt^2} -t\frac{dx}{dt} - ye^t =t^2$$
0
votes
1answer
12 views

Differential equations - maximal domain

I was solving an exercise about differential equations, and i really don't get how can I determinate the maximal domain of solution. Example: $$(dy/dx) = x - y/(1+x), y(0) =-1$$ The solution is ...
0
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0answers
18 views

Solving nonlinear system of ODEs

I have the following system of differential equations: $$ \begin{cases} \frac{dx}{dt} = (1 - y) x - 0.4 xu \\ \frac{dy}{dt} = (x - 1)y - 0.2yu \\ \psi_1' = - \frac{dH}{dx} = (-1 + 0.4u)\psi_1 + y ...
0
votes
1answer
19 views

mass-spring system. what is y(t)? [on hold]

Consider a mass-spring system with unit mass (m = 1), spring constant k = 9, critically damped, and no external force. Suppose that the oscillator starts at rest, and slightly compressed, at the point ...
0
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0answers
24 views

Inhomogeneous ODE (2nd order) - question to Laplace-transformation?

I've the following inhomogeneous second order ODE: $$a_1\cdot u(t) + a_2\cdot u'(t) + a_3\cdot u''(t) = b_1\cdot y(t) + b_2\cdot y'(t) + b_3\cdot y''(t)$$ The parameters $a_i$ and $b_i$ are ...
0
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2answers
27 views

After my first question I tried to solve the equation $\frac{dT_i}{dt}=\frac{1}{RC}(T_a-T_i)+\frac{1}{C} \Phi_h$

After my first question I tried to solve my differential equation $$\frac{dT_i}{dt}=\frac{1}{RC}(T_a-T_i)+\frac{1}{C} \Phi_h$$ Here is what I have done until now. I used $y'=b-a \cdot y$ and the ...
1
vote
3answers
54 views

Showing unstablity of differential equation.

Assume differential equation $$ x'=2x+y+x \cos t-y \sin t $$ $$ y'=-x+2y-x\cos t+y \sin t $$ Show that solution $(x(t),y(t))=(0,0)$ is unstable. Is there a non-trival solution such that ...
0
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0answers
16 views

continuous random walks, wiener process, ito process: “snowballing” for high enough volatility?

I'm finishing a project for my ODE class and ran into some strange behavior involving a SDE (not exactly sure how to say this, but...) generated by an Ito process, using the Wiener process. I guess ...
0
votes
1answer
17 views

How to express a system of differential equations in a form suitable for numerical methods?

I am modeling rocket thrust equations using some of the formulas and derivations on page 37 & 38 here. For my Rocket model, I have the following two equations: $$dv/dt = 383v^2$$ $$dA/dt = 635.14 ...
2
votes
1answer
57 views

Proving inequality $(x^2+y^2)(y-1)+yx-y^2<0$

I have an inequality which came out of Lyapunov function for system of ODE's: $$(x^2+y^2)(y-1)+yx-y^2<0.$$ To prove stability of my solution, I have to prove that the inequalty is true in area ...
4
votes
1answer
38 views

Can the Heat Equation be Averaged Over a Region?

I am doing a project for my partial differential equations class in which I am motivating the definition of a weak solution. To get started, I assumed that $T$ was a solution to $\nabla^2 T = \partial ...
0
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0answers
11 views

Logistic model. Did I set up the differential equation $(1)$ correctly?

Update: I fixed it. The major mistake I made was that originally put $I(t) = \beta\cdot(P-y(t))$ while it of course is supposed to be $I(t) = \beta\cdot y(t)$. NB: I came up with this problem ...
0
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0answers
24 views

$\Phi(t)=P(t)e^{tR}$ as a fundamental set for $x''(t)=\sin(t)x'(t)$

Problem. Find $2\times2$ matrices $R$ and $P(t)$ such that $R$ is constant, $P(t)$ is periodic, and $\Phi(t)=P(t)e^{tR}$ is a fundamental set of solutions for $x''(t)=\sin(t)x'(t)$. $ $ Attempt at ...
0
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0answers
11 views

Existence And Uniqueness Theorem Question

(a) Does the existence and uniqueness theorem guarantee the uniqueness of the solution of the initial value problem $dy/dx = 2x(y-2)^\frac{2}{3}, y(1) = 2$ Attempt: NO because $∂/∂y = \frac{4x}{3 ...
1
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3answers
30 views

How to separate variables in this equation: $\;y\frac{dy}{dx} = (x+7)(y^2+6)\;?$

I need to solve the differential equation $$y\frac{dy}{dx} = (x+7)(y^2+6)$$ I know that the first step is to isolate both term each side and then integrate... But I can't figure out how to isolate ...
1
vote
1answer
39 views

When integrating, can only one term of an equation be integrated or must entire equation be integrated to maintain equality?

Is integration considered a basic operation in the sense you have to do it to all parts of the equation? $y dy - x dx = 0$ Is it valid to do $\int y dy - \int x dx = \int 0$ but invalid to leave out ...
-1
votes
1answer
34 views

How to put the equation $y'' + ky =0$ into Sturm-Liouville form?

I just wondering how do you put $$y'' + ky =0$$ into Sturm-Liouville form. Reason: I am trying to determine if the equation is Sturm-Liouville on the interval $[-3,4]$.
0
votes
1answer
26 views

Power Series Solutions And Minimum Radius of convergence [on hold]

Help with power series and minimum radius of convergence. Does the equation $$ (x^2 + 25)y'' + xy' + x^3y = 0 $$ have a power series solution $y = \sum_{n=0}^\infty c_n x^n$? If yes, ...
3
votes
3answers
56 views

solve $y'=ay+b$ [duplicate]

I have this differential equation which I want to solve $\displaystyle\frac{dT_i}{dt}=\frac{1}{RC}(T_a-T_i)+\frac{1}{C} \Phi_h$ I know it is in the form $y'=ay+b$ But how can I solve it ?
0
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0answers
20 views

Initial conditions of a transformed differential equation

I'm not sure if the title is appropriate but this is what I mean. Suppose $$ \frac{dy}{dt} = y(t),~~~ y(t_0) = y_0~~~~~~~~~~~~~(1)$$ and $$ \frac{dx}{dt} = x(t),~~~x(t_0) = x_0 ~~~~~~~~~~~~~~(2)$$ ...
3
votes
3answers
72 views

Why before $e^{x}$,the solution was not possible?

we know the important role of exponential function in solving of ordinary differential equation.But the solution can be done by using another function like $10^x$ or $2^x$.The example below shows ...
1
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0answers
17 views

Phase Portrait of DE's

How would I graph the phase portrait of $$ x' = x^2+y^2-2 \qquad y' = y-x^2 $$ ? Could someone provide some insight by hand or perhaps a computer-generated image?
0
votes
2answers
32 views

Help with first order linear PDE with initial condition

I would like to solve the following pde: $2y\cdot \partial_x u(x,y)-3x\cdot\partial_yu(x,y)=0$ and $u(x,x)=e^{x^2}$ Without the initial condition I got the following result: ...
0
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0answers
17 views

Solve DE with trasfer function

How would you solve $\frac{\mathrm{d} C}{\mathrm{d} t}=\frac{F}{V}C_{i}-\frac{F}{V}C-kC^{3}$ given $\frac{F}{V}=0.1, k=0.5,$ $\frac{\mathrm{d} C}{\mathrm{d}}=0$ ? I´m supposed to use a transfer ...
1
vote
1answer
19 views

Why is it sometimes it seems like you can integrate with respect to x or y and treat the other as a constant, and other times you can't?

I am very confused right now. I thought we can't just do algebra on an ODE to find the solution. The following isn't allowed: $x dx + (y - 2x)dy = 0$ $x dx=-(y-2x)dy$ $\int x dx = \int -y+2x dy$ ...
1
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2answers
29 views

Getting wrong answer with exact equation with initial condition.

Solve the initial value problem $(4y+2t-5)dt+(6y+4t-1)dy=0, y(-1)=2$ This is an exact equation with $M(t,y)=\frac{\partial f}{\partial y}=4y+2t-5$ and $N(t,y)=\frac{\partial f}{\partial t}=6y+4t-1$ ...
2
votes
1answer
44 views

Differential Equation - Water evaporation

Given that a glass of water is filled to its fullest, $10\,cm$ in height, and that after three days the water level is at $9\,cm$ in height. Find when the glass will be empty. The water is ...
0
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0answers
7 views

Locally Linear Systems-repeated $\lambda$

For a locally system whose corresponding linear system has repeated eigenvalues, the type of equilibrium point cannot be determined. I know that the locally Linear system equilibrium can possibly be a ...
0
votes
2answers
36 views

How to find eigenvalues of this matrix

How to find eigenvalues of this matrix: $\left( \begin{array}{ c c } 2 & 0 & 0 \\ 0 & 2 & 4 \\ 0 & -1 & 2 \end{array} \right) $ ATTEMPT: $2-λ [(2-λ)(2-λ) ...
0
votes
1answer
31 views

Looking for a primitive …

The problem is to find $f$ such that $$f^{\prime}(x)+\int_0^x f(t)\times u(t)dt=0$$ where $u$ is given. I tried to find a primitive of the function $\frac{f^{\prime\prime}}{f}$ but I think it is not ...
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0answers
15 views

Find the general solution for the following non-homogeneous equation?

$y^{(4)} - 2y''' + y'' = 2e^{3x} + x$ Attempt: The characteristic equation is $r^4 - 2r^3 + r^2 = 0$. ==> $r^2 (r^2 - 2r + 1) = 0$ ==> $r^2 (r - 1)^2 = 0 $ ==> $r = 0, 0, 1, 1$. So, $$y_h = ...
1
vote
2answers
32 views

Solve $x''(t)-\frac{x^2(t)}{\sin t}=\frac{\sin\left( (t-1)^2\right)}{\sin t}$.

Solve the following Cauchy problem: $$x''(t)-\frac{x^2(t)}{\sin t}=\frac{\sin\left( (t-1)^2\right)}{\sin t}$$ with $x'(1)=x(1)=0$. I would appreciate some help with this problem. Thank you very ...
1
vote
2answers
56 views

solving differential equation second order

Can anyone please explain me step by step how to solve this differential equation: $$\begin{align*} y'' + w^2 y &= 0 \\ y(a) &= A \\ y'(a) &= B \end{align*}$$
0
votes
4answers
40 views

Differentiation of function to the power x

Given the function $f(x)=(2 + ln(x))^{x}$, find $f'(1$). This is what I tried: $$f(x)=(2 + \ln(x))^{x}=e^{2x+x\ln(x)}$$ So the derivative would be: $$f'(x)=(2x + ...
1
vote
2answers
48 views

Solving system of differential equations $\dot{x}=3x - 2y$, $\dot y = 2x - y + 15 e^t \sqrt{t}$

I am having problem with system \begin{cases} \dot{x}=3x - 2y;\\ \dot y = 2x - y + 15 e^t \sqrt{t}. \end{cases} Eigenvalues are $\lambda_1=\lambda_2=1$, the only eigenvector is $V_1 = (1,1)^T.$ I ...
0
votes
2answers
32 views

Differential equation problem. Integrating the logistic equation. [duplicate]

I would like to know how to integrate or rather solve this: $$ \frac{dP}{dt} = kP(L-P). $$ I have the solution, but I would like to know how to arrive at it. I have been told it involves separation ...
-1
votes
2answers
19 views

Create a non-linear first order differential equation which can be used using the method of separation. [on hold]

In addition, I will need to solve this equation and determine the interval of existence for the solution. Any suggestions?
2
votes
1answer
32 views

Solve a second order nonlinear equation

I have a second order nonlinear equation: $$-u''+ \frac{1}{4}(u')^2+au=x^2.$$ I am only interested in the solutions in $[0, \frac{x^2}{a}+\frac{1}{a^2}]$. One paper claims without proof that the ...
1
vote
0answers
28 views

Arnold ODE Problem

Problem 1 of Section 1.2.4 of Arnold's ODE book asks, "Can the integral curves of a smooth (continuously differentiable) equation $\frac{dx}{dt} = v(x)$ approach each other faster than exponentially ...
0
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0answers
10 views

detailed balance condition for coupled Langevin equation

Suppose $a$ and $m$ are real variables and they satisfy the following two coupled Langevin equations: $$ \dot{a}=F_a(a,m)+\eta_a(t);\quad\dot{m}=F_m(a,m)+\eta_m(t) $$ where $\eta_a$ and $\eta_m$ are ...
0
votes
1answer
41 views

Differential equation $(x^2y^2-1)dy+2xy^3dx=0$

$(x^2y^2-1)dy+2xy^3dx=0$ problem states that $y=t^n$ must be used. Using software it seems that there is a real solution. $$\frac{1}{3} \left(-\frac{\sqrt[3]{3 \sqrt{9 c_1^4-4 c_1^2 x^6}-9 c_1^2+2 ...
1
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0answers
67 views

Meaning of $dx$ [duplicate]

If I remember correctly, we use $ Δx$ for changes in $x$ and when $Δx \rightarrow 0$ then $ Δx$ takes the form of $dx$?