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

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1answer
16 views

Inverse Laplace Transformation of a heaviside function.

I'm working through an example of an inverse laplace transformation: $$\mathscr{L}^{-1}[\frac{e^{-3s}}{s+1}] = u_3(t)e^{-(t-3)}$$ I am having trouble seeing how this works. I know that: ...
1
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2answers
46 views

Finding solutions for the Riccati equation $y'+y+y^2 = 2$

Find two solutions in $\mathbb{R}$ for the Riccati equation $$y'+y+y^2 = 2$$ Which satisfy the following initial conditions: a) $y = b$ when $x=0$, $-2 \leq b < 1$ b) $y = b$ when ...
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0answers
57 views

How to understand Differential Equations?

I'm taking Differential Equations -II course this semestre and I'm in a big trouble. The problem is, I don't know how can we find the solutions, how can we be sure whether the solution(s) exist or ...
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0answers
15 views

Coupled second order differential equations

I need to solve $x''-ay'' -x(y')^2-b\cos(y)=0 \\ 2xx'y' + (a^2+x^2)y''-ax''+ab\cos(y)+bx\sin(y)=0$ where $x$ and $y$ are both functions of $t$. I'm not even sure where to start. How can I solve ...
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0answers
36 views

Differential Equation [closed]

I. Consider the differential equation $$(NH)\qquad y'''+16y' = 64 \tan 4x$$ and open interval $I = (−π/8, π/8)$. Find the following. A. (4) Find linearly independent solutions $y_1(x)$, $y_2(x)$, and ...
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0answers
25 views

Solution to a ODE system using a power series

I'm certain the pattern the system creates is $$ A^kX(0) = \begin{pmatrix}2^k\\1\\2^k\end{pmatrix}\hspace{3pc} $$ Where A is a matrix created by the system and X(0) is a solution vector at t=0 Im ...
2
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1answer
41 views

Solutions of the constant coefficient Helmholtz equation via the Fourier transform

When $f$ is a rapidly decaying Schwartz function, $$ g(x) = \frac{1}{2\lambda} \int_{\mathbb{R}} \sin \left(2\lambda\left|x-y\right|\right) f(y)\ dy $$ is an element of ...
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0answers
22 views

Solving this ODE through derivatives?

I'm currently trying to solve a more complex system of ODE. In order to understand the environment more, I've simplified stuff a bit, such that I have arrived at $$ \alpha F(x) = G(x) + ...
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0answers
20 views

How to solve matrix differential equations problem

I need to solve $$\begin{bmatrix} 20 & 6 \\ 6 & 7 \end{bmatrix} \begin{bmatrix} y''_1 \\ y''_2 \end{bmatrix} = \begin{bmatrix} 40 \\ 15 \end{bmatrix}$$ And I'm ...
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0answers
39 views

System of two ODE

I need to solve $$ \alpha F(x, y_0) = G(x) + y_0 + F_x(x, y_0)(H(x, F(x, y_0)) + \lambda_0 (F(x, y_{1}) - F(x, y_0)) \\ \alpha F(x, y_1) = G(x) + y_1 + F_x(x, y_1)(H(x, F(x, y_1)) + \lambda_1 (F(x, ...
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2answers
42 views

Solving $y'(x)-2xy(x)=2x$ by using power series

I have a first order differential equation: $y'(x)-2xy(x)=2x$ I want to construct a function that satisfies this equation by using power series. General approach: $y(x)=\sum_0^\infty a_nx^n$ ...
2
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2answers
35 views

Inverse Laplace Transform of $\ln[\frac{s^2+a^2}{s^2+b^2}]$

How does one find $\mathcal{L}^{-1}\{\ln[\frac{s^2+a^2}{s^2+b^2}]\}$? I've tried splitting it up into $\mathcal{L}^{-1}\{\ln(s^2+a^2)\}-\mathcal{L}^{-1}\{\ln(s^2+b^2)\}$. However, I can't think of ...
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0answers
19 views

Solving ODE ${dV \over dt} + \frac{1}{\tau} V = f(t) = Ae^{j \omega t }$

For solving ${dV \over dt} + \frac{1}{\tau} V = f(t) = Ae^{j \omega t }$, Wikipedia says the solution is $V_0e^{-t/\tau} +A\frac{1}{j\omega +1/\tau} \left( e^{j \omega t} - e^{-t/\tau}\right)$. But I ...
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1answer
31 views

when eigenfunction is orthogonal?explain with example plz [closed]

Please clear my concept I think eigenfunction is orthogonal when integral is zero but there is no idea for solving this. Please explain with example.
1
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1answer
27 views

Principle of superposition

Here $\phi$ is the solution to a linear pde so the principle of superposition applies. $\theta$ is the phase. I've tried using trig identities and different linear combinations in order to try and ...
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0answers
36 views

Find all the points on the level surface

Find all points on the level surface $$x^3 - 3x - y^2 + 4y + z^3 = 10$$ Where the tangent plane is horizontal My answer is $(1,2,2)$ and $(-1,2,4^{1/3})$. I don't know my answer is correct. Someone ...
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0answers
22 views

Solving a differential equation using the contraction principle

I am trying to solve this simple initial value problem from a differential equations textbook I have. Does solving the problem using the contraction principle simply mean showing that Picard's ...
3
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3answers
42 views

Laplace transforms for a pharmacokinetics multi-compartmental model

I am an anaesthetist trying to write some pharmacokinetics software as a pet project. Unfortunately the maths I need is a bit too much for my rusty high school calculus, and I am out of my depth. I am ...
2
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3answers
87 views

Nonlinear second-order ODE $yy'' - (y')^{2} = y^4$

I have the following ODE to solve. $$ yy'' - (y')^{2} = y^4 $$ I tried to substitute $y'$ by $v$, and then I get the following: $$ yv' - v^{2} = y^4. $$ I can't go further. I can't see what I'm ...
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2answers
44 views

Fourier inverse of a function to get dirac

I'm trying to get the dirac function from a fourier inverse tranform: $$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iw(x-x_0)}dw$$ It is this last step I am stuck on to get the conclusion. Original ...
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1answer
24 views

Solving $x(x+1)y' + y = x(x+1)^{2}e^{-x^2}$ in $(-1,0)$

What I tried to do: I divided both sides by $x(x+1)$: $$y' + \frac{1}{x(x+1)}y = (x+1)e^{-x^2}$$ This has the form $$y'+P(x)y = Q(x)$$ The general solution would be (supposing $f(a)=b, a \in (-1,0)$ ...
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1answer
39 views

How to solve a system of linear equations exactly?

Given a system of equations, $$\frac{dR}{dt}=-aR+bJ, \quad \frac{dJ}{dt}=-aJ+bR,$$ I have to discuss what happens to their love(Romeo and Juliet mathematical modeling exercise) is their caution $a$ ...
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0answers
37 views

Maple, solve coupled differential equation

I'm having trouble getting maple to dsolve this coupled differential equation. When I try to solve it, it gets stuck in evaluation and freezes. Is it possible to get an answer as a real decimal ...
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1answer
19 views

Numerical solution 1st order ODE with Euler's method

I'm trying to solve this 1st order ODE numerically by bringing it into an explicit form, but I don't think it is valid because of the dependency on x_n in the final expression. $$ \frac{d y}{d x} + x ...
2
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0answers
19 views

ODE from systems biology, can I generalize this? Have solution but not sure how to arrive at it.

Reading a systems biology book, and it describes a model with the following ODE: $$ \frac{dY}{dt} = -\gamma Y + v_1 X_1 (T - Y) + v_2 X_2 (T - Y)$$ where $Y$, $T$, $T - Y$, $X_i$, $a$ and $v_i$ are ...
1
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1answer
28 views

Uniqueness of the Greens function

I'm a little confused about the interpretation of the Greens function. My understanding is that the Greens function is related to the solution of differential equations, so it should be unique. For a ...
2
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2answers
42 views

Differentiating exponential functions - is base e the only situation?

My maths book gives the example of; Where $$ f = e^x $$ $$ f` = e^x $$ It only uses the example of base e in all of the questions so does that mean this is the only situation where the differential ...
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0answers
16 views

nonlinear version of strong maximum principle

page 31 Vazquez has a nonlinear version of strong maximum principle which designated for Quasilinear parabolic PDE. Roughly it says, if $u_0\le v_0$, then either we have $u\equiv v$ or $u<v$ for ...
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1answer
29 views

How should I go about obtaining the explicit solution to this logistic first-order nonlinear ordinary differential equation?

I have to find the explicit solution to this harvesting problem in a population model where $\frac{dN}{dt}=rN(1-\frac{N}{K})-H(N)$ such that $H=qEN$, subject to initial condition $N(0)=N_0$. Here ...
0
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1answer
28 views

Applying boundary conditins to a differential equation confusion [closed]

Here $k, A_1, A_2$ are constants Although $A_1$cos$kx+A_2$sin$kx \not\equiv 0$, $A_1$cos$kx+A_2$sin$kx$ can equal zero at certain values of $x$ For example if $A_1=1, A_2=1, k=1$ then ...
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0answers
26 views

find the fourier transform of $xf(x)$ appended

I've seen the method in which you prove this fourier transform, but what if you don't recognize that $$xf(x) e^{i k x} = \frac{1}{i} \frac{\partial}{\partial k} \Big[ f(x) e^{i k x} \Big] $$ would I ...
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1answer
44 views

Solving Eikonal Equation

The problem is the following: I have the bidimensional eikonal equation with non-constant propagation: $u_x^2+u_y^2=u^2$ The goal is: i) To find the characteristic strips for the parametrization ...
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0answers
27 views

A few questions about eignenvectors and the associated root vectors.

Let A be the matrix formed from the original system of equations and t is a repeated eigenvalue. I've noticed when solving problems containing eigenvectors of multiplicty >1 that when the ...
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0answers
27 views

Hopf bifurcation in linear delay differential equations?

Can equilibria of a system of linear delay differential equations undergo a Hopf bifurcation? I am able to generate oscillatory solutions by tuning a time delay in the system, but would it make sense ...
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2answers
38 views

ordinary differential equation solution for an example [closed]

I can not understand why $\frac{dx}{dt}=cx(t)$ has as solution $x(t)=x(0)\cdot e^{ct}$? Thank you very much for the help.
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1answer
24 views

Clarification about second order Linear ODE with constant coefficients

I'm studying the steps to obtain the general solution of a second order Linear ODE with constant coefficients, but I haven't understood a justification. $$y''+ay'+by=f$$ Let's look for a solution ...
1
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1answer
11 views

Show behavior of Fourier Transform

If F(w) is the Fourier transform of f(x), show that F(aw) is the Fourier transform of (1/a)f(x/a). So if I apply a fourier transform to (1/a)f(x/a): $$ \frac{1}{2\pi}\int_{-\infty}^\infty ...
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0answers
15 views

existence of Laplace transform interval [closed]

Does the following Laplace transform exist? $\mathcal{L}\{ (-t-1)^{1/2}\}$ This function is defined on $(-\infty, -1]$ and therefore does not comply the definition: $f(t)$ for $t \geq 0$.
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0answers
15 views

Particular solution of hypergeometric differential equation with inhomogeneous term

I'm looking for the particular solution of an nonhomogeneous second order differential equation. The inhomogeneous term is a function of the form: $\frac{1}{u}$. I've tried to find the particular ...
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0answers
25 views

Find the radius of convergence of solution's series of differential equation

In general how can I Find the radius of convergence of solution's series of differential equation . In particular if we have the following differential equations $$ (1-x^2)y''-2xy'+λ(λ+1)y=0$$ And ...
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2answers
22 views

Dimension of the space of solutions to the differential equation $x' = Ax$

I have no idea where to begin this proof. Any help would be greatly appreciated!
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1answer
24 views

Using Phase planes, how do I find graphically the equilibria and their stability of a logistic growth model??

I'm having trouble understanding the concept of phase portrait which I never learned in my applied differential equations class. The question is asking to study the logistic growth model, $$ ...
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1answer
20 views

Find the solution to variable seperable differential equation [closed]

Find the solution to the differential equation $\frac{dy}{dx}=\left(4x^3+12x^5\right)e^{-y}$ that satisfies that condition $y(1) = \ln5$.
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0answers
21 views

Runge kutta 4th order computation of force solving 2nd order ode

\begin{equation} \frac{dx}{dt} = v \end{equation} \begin{equation} m .\frac{dv}{dt}= F_{p }(x)+F_{g}(v,x) \end{equation} Conside I am solving the above two equations using runge kutta 4th order ...
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0answers
24 views

Can't a diffrential equation have same particular solutions analogous to polynomial roots

Our professor introduced the solution of a diffrential equations analogous to polynomials. (Finding roots vs finding functions which satisfied a particular set of operations). While solving 2nd order ...
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0answers
227 views

For what values of t, the solution for this equation exist

I need help in finding maximal solution for the problem: $$ \cases {{\dot{x} = x^2+t}\\{x(0)=0}}$$ I know that because $x(1) \geq \frac{1}{2}$ and that every solution $x(t)$ of the problem is greater ...
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1answer
18 views

State space representation involving derivatives of input

We have the system $y''=-7y'-12y-u'-2u$ If we choose $x_1=y,x_2=y'$ we can write the system as $x'=Ax + Bu \\ y= Cx$ Finding A is easy, but how do I find expressions for $B$ and $C$ when we have ...
5
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1answer
82 views

Find $f(x) $ given that: $f'(x)=\frac{f(x)-x}{f(x)+x}$ [closed]

I would appreciate if somebody could help me with the following problem: Find $f(x)$ given that: $f \colon \mathbb{R^+} \rightarrow \mathbb{R^+}$, $f$ is differentiable function, and ...
1
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2answers
41 views

Applying Chain rule to $z = z(u, v) = f(x(u, v), y(u, v))$.

If $z = z(u, v) = f(x(u, v), y(u, v))$ is a differentiable function, where $x = x(u, v) = u^2 − v^2$, $y = y(u, v) = 2uv$, show that $$\frac{∂^2f}{∂x^2} +\frac{∂^2f}{∂y^2} =\frac{1}{4(u^2 + ...
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votes
2answers
23 views

finding the maximum of a second derivative of a function given its differential equation

Say I have $u(x)$ satisfying $\frac{d}{dx}(a(x)u'(x)) = 0$ with $u(0) = 0$ and $u(l) = 1$. Is there any way to find the maximum of $u''$, for example, without having to solve for the function $u$ ...