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

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1answer
11 views

What does the linearity of a derivative mean?

What does the linearity of the nth order derivatives of dependent variable y with respect to,say time t,means?
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2answers
46 views

Differential equation $f'''(x)=-f(x)$ with restriction using power series

Using power series, Prove the existence of a $C^3$ function (continuously differentiable 3 times) $f:\mathbb{R} \to \mathbb{R}$ such that $f'''(x)=-f(x)$ $\forall x \in \mathbb{R}$ and ...
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0answers
7 views

Finding inhomogeneous solution given homogeneous solution.

This is a practice problem for an upcoming exam. I know how to solve a problem like this when the solutions are not vectors, however, this is throwing me off quite a bit. I assume that $x^{(2)}$(t) ...
2
votes
4answers
45 views

How can I find an ODE equation from $dy/dx$

What is the ODE satisfied by $y=y(x)$ given that $$\frac{dy}{dx} = \frac{-x-2y}{y-2x}$$ I understand that I need to get it in some form of $\int \cdots \;dy = \int \cdots \; dx$, but am not sure ...
2
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3answers
42 views

Classifying a differential equation

How do I classify the following differential equation? In particular, is this differential equation "homogeneous?" $$(x^3+3y^2)dx-2xydy=0$$ Solving it is not the problem, but I don't know how ...
2
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5answers
34 views

Finding a general solution to a differential equation, using the integration factor method

Use the method of integrating factor to solve the linear ODE $$ y' + 2xy = e^{−x^2}.$$ And verify your answer I can solve the ODE as a linear equation (mulitply both sides, subsititute, reverse ...
3
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1answer
35 views

From Gravity Equation-of-Motion to General Solution in Polar Coordinates

I'm having trouble getting the general solution of this differential equation. The gravitational equation of motion is, for constants $M$ and $G$ and position vector $\vec{r}$, ...
0
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1answer
24 views

How to find the solution to the differential equation $dy/dx$

How do I find the solution to the differential equation: $ \frac{dy}{dx} = e^{α(x+y)} + 3e^{αy}$? where $\alpha$ is not zero.
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3answers
10 views

rewritten the Gompertz finction

Why can I rewrite this equation: $$\frac{dN}{dt}=rN ln(\frac{K}{N})$$ as $$\frac{d}{dt}[ln(\frac{N}{K} )] = r(-1)[ln(\frac{N}{K})] $$ I have tried: I know that since I can do ...
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1answer
55 views

How to solve the differential equation $y^3y''+1=0$ [on hold]

I do not know how to solve: $y^3y''+1=0$ $2yy''-3(y')^2=4y^2$ $(y-1)y''=2(y')^2$ My answers are not the same as in the book. The answers should be: 1. $(sqrt(Cy^2+1))/C=D+2t$ 2. there are two ...
-1
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0answers
61 views

If $f'(ax) = \frac{b}{a}$ then $\frac{\textrm{d} x}{\textrm{d} a} > 0$ where$f, f' > 0$ and $f'' < 0$?

I'd appreciate any help I could get for this question: Define $f:[0, +\infty) \rightarrow R^+$ with $f(0) = 0$ and $f(\infty) = \infty$, and such that $f' > 0$ and $f'' < 0$. I have the ...
0
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1answer
27 views

Can a differential equation be both homogeneous differential equation and separable differential equation?

This tutorial of Khan Academy suggests that if dy/dx = f(x,y) can be written as ...
1
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3answers
29 views

How to verify my solution to an separable differential equation?

I have this question: Find the general solution to the separable differential equation $$ \frac{dy}{dx} = y(1-y). $$ My attempt is : $$ \frac{dy}{y(1-y)} = dx $$ $$ \frac{1}{y(1-y)} = ...
2
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1answer
42 views

Finding a Solution to a linear Voltera equation of the second type

I want to solve the following integral equation: $$ u(t) = \int_t^T a(s) ds + \int_t^T b(s)u(s) ds , $$ with $a, b, u$ being functions from $[t,T] \rightarrow \mathbf{R} $. I transformed the ...
2
votes
3answers
43 views

Substitution to solve an initial value problem

By using the substitution $y(x) = v(x)x$, how can I solve the initial value problem $$ \frac{dy}{dx} = \frac{x^2+y^2}{xy - x^2},\quad y(1)=1 $$ And also keep my answer in the form $g(x,y)= 4e^{-1} ...
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0answers
14 views

A pde that cannot solves by Lax-Milgram theorem

Consider the following pde: $-u''(x)+au'(x)+bu(x)=f(x) \qquad\text{in}\; (0,1)\\$ $u'(0)=\alpha\\$ $u'(1)+u(1)=\beta$ How could I prove that it has a nontrivial solution? The bilinear form ...
1
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0answers
15 views

Inequality in a Dirichlet BVP

For the Drichlet boundary value problem $Lu = -u''+p(x)u'+q(x)u = f(x), \; \; \; x \in I=[a,b]$. with $u(a)=u(b)=0$. Then for $v \in H^2(I) \cap H_0^1(I)$ show that $\left\lvert \right\rvert v ...
1
vote
1answer
40 views

Partial D.E. Change of Variables

Can anyone tell me how to derive the value for alpha and beta, I'm guessing 6 and 1 in one order or another - "via quadratic". In transforming the equation, could someone show also how the operator ...
2
votes
1answer
49 views

Equilibrium Points Second Order Differential

Attempt: I get the system of the two first order equations (first order in $w$) by considering the different signs the first derivative takes. Problem is by equilibrium points: do I just set the ...
1
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1answer
35 views

ODE $x'' + 2x' +5x = \sin3t$, $x(0)=1,\ x'(0)=-1$, Solve using Laplace Transform

While solving the differential equation $$x'' + 2 x' + 5 x = \sin3t, \quad x(0) = 1, \quad x'(0) = -1$$ by use of Laplace transform I got to $$X(s^2 +2s+5)=\frac{(3)}{s^2 +9} +s +1$$ ...
3
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1answer
29 views

Poisson Process Derivation.

I was looking at a derivation for the poisson process , which tells the number of events occurring in time $t$ , I came across the following differential equation : $\frac{d}{dt}(P_n(t))$ = ...
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2answers
51 views

Do the initial value problem $dy/dx=2y^{1/2}$ , y(0)=a has infinitely many solution for a=0. [duplicate]

please solve the initial value problem $dy/dx=2y^{1/2}$, y(0)=a. I wanted to know that this problem admits infinitely many solutions for a=0 or admits infinitely many solutions for $a\geq 0$
2
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1answer
27 views

Basic Ordinary Differential Equation Help

I have the following $3$rd order, non-linear, homogeneous differential equation: $$y''' + ey'' + y' + (d + e)y - dy^2 = 0,$$ where $d,e$ are constants. My questions: Have I classified this ...
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2answers
39 views

Solving differential equation , Calculate concentration of sands in water?

$V=$ volume of water in the water tank $M(t) =$ mass of sands in the water at time $t$ $K(t) =$ Concentration of sands in water at time $t$ $R=$ rate of water flowing out Concentration sands in ...
4
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2answers
57 views

Existence of a solution for a nonlinear ODE on $[0,\infty)$

I'd like to prove that the solution to the following IVP exists on $[0,\infty)$. The IVP is given by $$ \begin{cases} y'(t) = y^2 \cos(t)-ye^t \\ y(0)= y_0 \end{cases} $$ where $y_0 ...
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0answers
14 views

PDE reduced to ODE Uniqueness??

Could you please help me with the following problem. As a first help, I know the solution of the following ODE: \begin{align} j_1(t)[r \log(j_1(t)) + \beta] &= j_1'(t) \\ \nonumber j_1(T) ...
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0answers
12 views

Quantative Methods for Business question [on hold]

Economists theorize that the recent recession has affected men more than women because men are typically employed in industries that have been hit hardest by the recession.​ Women, on the other​ hand, ...
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0answers
19 views

Existence and uniqueness solution of a differential equation

If I have the following equation: $\frac{\delta}{\delta t}y(t,r)=\int_0^1 G(|r-r'|)y(t,r')dr'e^{\int_0^t\int_0^1G(|r-r'|)y(s,r')dr'ds}-y(t,r)$ $ y(0,r)=a(r)$ where $G:\mathbb{R}^+\to\mathbb{R}$ is ...
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0answers
37 views

More elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$

Is there a more elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$ than to use Euler's identity and get the general solution through brute computation?
3
votes
1answer
29 views

Trajectories that connect equilibrium points

Suppose I consider the autonomous system \begin{align*} x' &= F(x, y)\\ y' &= G(x, y) \end{align*} where $F$ and $G$ are nonlinear and my task is to draw the phase portrait of the above ...
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1answer
44 views

Solve the differential equation $\{xy\log \frac{x}{y}\}dx+\{y^2-x^2 \log \frac{x}{y}\}dy=0$ given that $y(1)=0$.

Solve the differential equation $\{xy\log \frac{x}{y}\}dx+\{y^2-x^2 \log \frac{x}{y}\}dy=0$ given that $y(1)=0$. I was able to find the solution to it by the method of solving homogeneous ...
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0answers
18 views

Does solution to particular system of O.D.E. with boundary conditions exist and unique?

I am dealing with the following system of $2$nd order differential equations: $$\ddot{y_k}+\frac{\ddot{y}_{N,k}}{2}+a\dot{y}_{N,k} = \gamma_k^2 y_k$$ where $y_{N,k} = \sum_{i=1,i\not=k}^{N} y_i ...
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1answer
38 views

Solving differential equation using Laplace transform

Can this DE be solved using Laplace transform? $\frac{\mathrm{d} y}{\mathrm{d} x}\cos x=y\sin x+\cos ^{2}x$
1
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1answer
20 views

Changing a heaviside function into a one line function

$$h(t) = \left\{\begin{array}{l}1,\, \pi\leq t<2\pi\\ 0,\, 0\leq t<\pi\text{ and }t\geq2\pi\end{array}\right.$$ I need to change $h(t)$ into a one line function. I believe it to be ...
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1answer
20 views

How to calculate Z when doing Bernoulli differential equation?

I'm just learning how to do a Bernoulli differential equation and I'm stuck at the part where you have to use Z (others call it U). For example: When (y^-3)y' + (1/2x)y^-2 = -(1/2)X² * sin²x*cosx ...
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0answers
32 views

2nd Order Nonhomogeneous with varying coefficients

Is there a way to solve or get an analytical approximation to this equation? $z''(t) + z'(t)\frac{(\omega_0 + \Delta\omega (1 - e^{-\frac{t}{\tau }}))}{Q} + z(t)(\omega_0 + \Delta\omega (1 - ...
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2answers
44 views

Fitting driven Harmonic Oscillator

I've got some datapoints of a turning disc. It is supposed to obby the following differential equation: $I\ddot{\theta}+\gamma\dot{\theta}+k\theta=\tau$, So it should have the form of a driven ...
0
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1answer
27 views

How to differentiate with respect to component of a vector?

Let $\vec{\alpha}=\frac{m(\vec{x})}{x^2}\vec{x}$ where $\vec{x}=(x_1,\,x_2)$. In a book I read in Eq.(3.24), it was given that $$ \frac{\partial \alpha_1}{\partial x_1}=\frac{d m}{d ...
5
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0answers
36 views

Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$ f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
0
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1answer
36 views

Solve differential equation

How can we solve (if a closed form expression for f(x) can be found) the following first-order linear differential equation? $$f'(x)=f(x)\cdot (\cos x+\tan x)$$ I have found that one function which ...
2
votes
2answers
63 views

Solving $y' + \frac{1}{2}xy + y^{2} = 0$

I am trying to solve the ODE $$y' + \frac{1}{2}xy + y^{2} = 0.$$ Mathematica gives that the answer is $$y(x) = \frac{e^{-x^2/4}}{C + 2\int_{0}^{x/2}e^{-t^{2}}\, dt}.$$ Of course, if I take this answer ...
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2answers
18 views

Quadratic equation with several variables

How does $$y^{2} - 4y -t^{2} - C = 0$$ Become $$y = 2 \pm \sqrt{t^{2} +2C + 4}$$ I know its the quadratic formula but I dont know how it got it that point The original equation is $$\frac{dy}{dt} ...
3
votes
3answers
47 views

Laplace Transform of a Heaviside function

Find the Laplace transform. $$g(t)= (t-1) u_1(t) - 2(t-2) u_2(t) + (t-3) u_3(t)$$ I understand that the $\mathcal{L}\{u_c(t) f(t-c)\} = e^{-cs}*F(s)$ Finding $F(s)$ is the hard part for me. My ...
3
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4answers
92 views

Simple differential equation( introduction but need some basic explanation)

I have a couple of questions before I dig deeper into my calculus book. First: I have learned that $\frac{d}{dx}\frac{x}{y}$=$\frac{y x'-x y'}{y^2}$ never really gotten a proper explanation for ...
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2answers
72 views

Is $ \cos² y = 0 $ a solution?

I'm studying math for school. We're solving separable differential equations. One of the exercises is: $$ \frac{\Bbb d y}{\Bbb d x} = \frac{ (\cos y)^2 \tan y }{1+x²}$$ If you separate the ...
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0answers
18 views

How to find the matrix associated with the differential equation? [closed]

How to solve ordinary differential equations using matrices.
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0answers
21 views

Is there a physical meaning of ranking in differential algebra?

The main stone in the Ritt's Algorithm from differential algebra is ranking. If we consider an example of a differential polynomial with two variables $x$ and $y$. Then how can we say $x$ is ranked ...
1
vote
2answers
49 views

Differential equation: $Ay'' + By' + Cy = h(x)$

I'm stuck solving the equation $y'' - 3y' + 2y = 2x^3-30$. The auxiliary equation is $k^2 - 3k + 2 = 0$ where $k_1 = 1, k_2=3$. Thus the general solution is: $$y_g = C_1e^x + C_2e^{3x}$$ Then, I ...
1
vote
3answers
39 views

Modelling interest with differential equations (Interpretation)

I am having trouble interpreting the meaning of this differential equation model for interest on an account. The problem is as follows: Assume you have a bank account that grows at an annual ...
3
votes
1answer
59 views

PDE: solving Fokker-Planck equation with initial and boundary condition

Here is the problem. We have the following simple PDE: \begin{equation} \frac{\partial p(x,t)}{\partial t}= - a\frac{\partial p(x,t)}{\partial x} + \frac{D}{2} \frac{ \partial^2 p(x,t) }{\partial ...