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

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3
votes
2answers
27 views

Trouble solving this differential equation: $x'=3(x-2)$, $x(0)=-1$.

Find the solution of the differential equation x'=3(x-2) given initial value condition of x(0)=-1 Here's my attempt. x'=3(x-2) dx/dt = 3(x-2) dx/x-2 = 3dt int dx/x-2 = int 3dt+c ln|x-2| = 3 + C ...
3
votes
2answers
621 views

Spring Calculation - find mass

A spring with an -kg mass and a damping constant 9 can be held stretched 2.5 meters beyond its natural length by a force of 7.5 newtons. If the spring is stretched 5 meters beyond its natural length ...
0
votes
3answers
36 views

Having trouble verifying a solution for a differential equation

Verify that $x=(t+1)e^{2t}$ is a solution for $$x = 2x+e^{2t},\ \ x(0)=1$$ My approach so far is. $$x' =2x+e^{2t}$$ $$dx/dt = 2x+e^{2t}$$ $$\int(dx-2x) =\int e^{2t}dt + C$$ $$-x^2 = e^{2t}/2 + C$$ ...
0
votes
0answers
9 views

Integrate multi-variable autonomous ordinary differential equations using Runge Kutta 4

I have a first-order ordinary differential equation (ODE) of the form: $$ \mathbf{\dot{y} = A\cdot y+B\cdot u} $$ where $\mathbf{y}$, the state variable, is a $7\times 1$ vector; $\mathbf{u}$, the ...
0
votes
0answers
12 views

Is this end-point map surjective

Consider the differential equation: $\frac{d U_s}{dt} = (a + w(s)b)U_s$ where $w$ is some unknown, smooth, real and bounded function on the interval $[0,T]$ and $a,b \in \mathfrak{su}(n)$. Let ...
2
votes
1answer
3k views

Solve a second order DEQ using Euler's method in MATLAB

I need to solve the equation below with Euler's method: $$y''+ \pi ye^{x/3}(2y' \sin(\pi x)+\pi y\cos (\pi x)) = \frac{y}{9}$$ for the initial conditions $y(0)=1$, $y'(0)=-1/3$ So I know I ...
2
votes
1answer
22 views

Find piecewise constant function u for $X'(t)=AX(t) + Bu(t)$ and $X(t)=\begin{pmatrix}10 \\0 \end{pmatrix}$ for some T

Consider the system $$x''(t)=u(t)$$ such that $x(0)=100, \; x'(0)=50$. Find a function $u$ piecewise constant such that $x(T)=0, \; x'(T)=10$ for a time $T$ Using the control theory language, it is ...
2
votes
1answer
76 views

Is okay to have different solution to differential equation?

Suppose I have the following differential equation: $ydx - xdy - dx = 0$ Now, I could divide it by Integrating factor $x^2$ to get: $(xdy - ydx)/(x^2) - dx/x^2 = 0$ Use the inspection rule to get: ...
3
votes
1answer
33 views
+50

Showing a bound exists

I was able to derive the following differential equations I have to work with for a function $V$: $$ \begin{align*} dV(x_1,x_2,x_3,x_4) &= ...
2
votes
1answer
57 views

Solve one dimensional wave equation using fourier transform

I'm trying solve this wave equation using Fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
-1
votes
0answers
34 views

Second-order nonlinear differential equation

I am trying to solve the following differential equation: $ \ddot{x}(t) + a\ |\dot{x}(t)|^n\ sign(\dot{x}(t)) + b\ x(t) = c\ sin(\omega\ t) $ where $n$, $a$, $b$, $c$, $\omega$ are constants, ...
1
vote
2answers
34 views

ODE using Laplace transform

[ I got my Y(t) to be : $$12 \, e^{-4} \, e^{-2s} \, [\frac{1}{12(s+2)} + \frac{1}{4(s-2)} - \frac{1}{3(s-1)}] + \frac{1}{(s-2)} - \frac{1}{(s-1)}.$$ so i assume I need to use t shifting for the ...
0
votes
0answers
11 views

Solving a system of ODEs with 4 repeated eigenvalues

I'm working on problem which requires me to solve a system of ODEs with 7 equations. I've gotten as far as determining the eigenvalues and vectors of my coefficient matrix $A$, but 4 of the ...
2
votes
0answers
15 views

Tough NL Diff Eq.

I'm trying to explore $$ \left( y'' + (1/x) \, y' \right)(1-y) \, – \, (1/x)\left(y'\right)^4 = 0 $$ with the initial conditions $y(0) = 0$ and $y'(0) = 1$. By substitution I can show that an ...
0
votes
0answers
18 views

*Solved* Terminology in DE, difference between Particular and Actual solution

Yesterday I started studying and preparing for a course in Differential Equations and today I came across something that confuses me; I watched a lecture on IVP and they used both Actual solution and ...
3
votes
0answers
35 views

Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
0
votes
0answers
36 views

Issue in first order differential equation

I've tried many times to reach the solution of a first order differential equation (of the last equation) but unfortunately I couldn't. Could you please help me to know how did he get this solution. ...
0
votes
0answers
25 views

In initial value second order DE problem, should the 2 conditions be at the same $x_0$?

Let's say that I have DE of $y''+p(x)y'+q(x)y=0$. To pick a particular solution, should the two conditions be [$y(x_0)=k_1$ and $y'(x_0)=k_2$]? or can be any other combinations of: [$y(x_0)=k_1$ and ...
0
votes
0answers
23 views

Differential Operator simplifying

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" by Elna Browning McBride In Sec 5" The extended form of the group generated by B and C " I did not understand ...
1
vote
2answers
42 views

Solving Simple Partial Differential Equation

I don't remember how i can solve this simple partial differential equation. Can someone help me? $$x\frac{\partial \phi}{\partial x}+y\frac{\partial \phi}{\partial y}+ (\alpha+1-x)\phi =0$$ Update ...
1
vote
2answers
39 views

$y'=\frac{y^2}{2x(y-x)}$

I'm trying to solve the following differential equation: $$y'=\frac{y^2}{2x(y-x)}$$ It is supposed to have a relatively easy general solution, but I can't find it. I've tried several things, the ...
-1
votes
0answers
23 views

Problem with initial values ODE

EQ = $y'+2xy=x$ Initial Value=$y(0)=-2$ $y'+2xy=x$ = $y'+y = \frac{1}{2}$ The solution of the Diff Equation $\frac{1}{e^x}$ $\int{\frac{1}{2}}e^xdx$ = $\frac{1}{2}+c$ I wonder how to check if this ...
4
votes
3answers
185 views

Differential algebra and differential-algebraic equations

Could you give me some information about differential algebra? What is it about? Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, ...
-1
votes
0answers
15 views

Use the lemma in this section to show that if T is an invertible linear transformation

Use the lemma in this section to show that if T is an invertible linear transformation then ||T||> 0 and ||T^-1|| is greater than or equal to 1/||T||. Lemma: For S, T in L(ℝ) and x in ℝ 1.|T(x)|is ...
2
votes
2answers
67 views

How do I find the exact solution to the boundary value problem $y'' = 4y' + y + 2 − 8x − x^ {2}$ , $y(0) = 0$ and $ y(4) = 16$?

I am approaching this question by trying to guess the general solution to the boundary value problem. However I haven't come up with one. Can someone explain how to solve this question please?
3
votes
2answers
130 views

How to apply the Gronwall lemma

Consider $x'=f(x)$ such that $(x_1,x_2)\mapsto(-x_1+2x_2,-2x_1-x_2)$. Show that for two solutions $x(t)$ and $y(t)$ of the above differential equation, we have: $$\lVert x(t)-y(t)\rVert \leq ...
0
votes
1answer
26 views
0
votes
1answer
32 views

matrix differential equation and its stability

I have a differential equation of a $n\times n$ real matrix $X$: $$\dot{X}=-AX$$ $A$ is also a $n\times n$ real matrix. Two questions: 1) What conditions should $A$ satisfy if we want that $X=0$ be ...
1
vote
0answers
14 views

Conformal mapping for constant Gauss Curvature

The Sine-Gordon equation describes varying angles, conserving differential lengths in a mapping with constant Gauss curvature by means of an ODE. In which conformal mapping (conserving angles), can ...
4
votes
2answers
129 views

Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
2
votes
1answer
45 views

Solving Bessel's ODE problem with Green's Function

If we have an inhomogeneous boundary value problem $x^2 y'' + xy' + (x^2 -1)y = x,$ $y(0) = y(b) = 0,$ where $b>0$ How to use Green's Funtion to Solve this problem. I am facing issues with ...
0
votes
1answer
28 views

Solution of $xu_x + yu_y = 0$

I have the first oder PDE $$xu_x + yu_y = 0 \; \text{on} \; \mathbb{R}^2$$ and I found the solution of that PDE is $$u(x,y) = f\left(\frac{y}{x}\right) = e^C = K$$ which is a constant solution. So, ...
0
votes
1answer
27 views

analytical solution of a nonlinear differential equation

can we find a closed form solution -- such as a series solution -- of the following equation $$\frac{dy_0}{dt}+b\left(\frac{20}{27}y_0(t)^2+\frac{10}{27}y_0(t)-\frac5{81} y_0(t)^3-\frac4{81}\right) ...
0
votes
1answer
23 views

Problem with initial values (Differential equations)

So i'm trying to solve a trivial problem but sadly I'm not good with math and i need help. SO I solve this equation $y'+y=2$ the solution was $2$, and the initial value $y(0)=2$. How can I check ...
2
votes
2answers
34 views

Discrete time equivalent to ODE

I'm reading a paper in which it is noted that $$\frac{dv(t)}{dt} = f(t) - \varepsilon v(t)$$ has the discrete time equivalent $$v(t+1) = v(t)\exp(-\varepsilon) + \frac{f(t)}{\varepsilon}[1 - ...
1
vote
1answer
295 views

How can i convert nonhomogeneous ode to homogeneous ?

I have an equation system $$y'(t) = M(t)y(t)+h(t)$$ where $[M(t)]_{2\times2}$ square matrix and $[h(t)]_{2 \times1}$ is the nonhomogeneous part of the system. I can solve numerically homogeneous ...
0
votes
1answer
33 views

differential equations solvable only by numerical methods [on hold]

What kind (a general formula would be nice) of differential equations do not have solutions expressible explicitly or implicitly or by an integral sign? In other words, what kind of differential ...
0
votes
1answer
34 views

Finding the differential equation, given a solution

I am unable to understand how to find the differential equation when a general solution has been given. Here are a few example solutions, which require their differential equations to be found: (a) ...
0
votes
1answer
15 views

Constant solutions and uniquenss of solutions theorem for IVPs

What role do constant solutions play in the existance and uniqueness theorem? For instance, consider the IVP $$\frac{dy}{dx} = x$$ $$ y(0) = 0 $$ Clearly, this IVP has a solution in the form of $y ...
0
votes
1answer
26 views

Exact differential equation problem

I was finding the solution of a differential equation. But I'm stuck on this part. I tried simple integration but answer is incorrect. I don't know how to solve this. $$ dz=(6x+3y)dx+(3x-4y)dy $$
0
votes
0answers
58 views

Integrating Factor. [duplicate]

$(axy^2 + by) dx + (bx^2y + ax) dy=0$ I have asked this question before too, but i wish to know the method for evaluating the integrating factor which is $\frac {1} {(a-b)(x^2y^2-xy)}.$ So far i ...
1
vote
0answers
10 views

Singular points while differentiating a function with respect to another function

I have $z(x) = \frac{df(x)}{dx}$ where $f(x)$ if a function of x. I'd like to have the derivative of $z(x)$ in respect to $f$: $\frac{dz}{df} = \frac{\partial f'(x)}{\partial x} \frac{dx}{df}$ ...
1
vote
1answer
33 views

Simple Harmonic Motion under Periodic disturbing force

A particle of mass $m$ is executing a SHM in a straight line under an acceleration $n^2 \times (distance)$. If a periodic force $mk \cos{pt}$ be introduced and the time period of forced vibration ...
4
votes
1answer
47 views

Still getting wrong answer after trying to solve $x''(t)+4x(t)=t^2$ where $x(0)=1$ and $x'(0)=2$

I am trying to solve this differential equation: $$x''(t)+4x(t)=t^2,x(0)=1,x'(0)=2$$ The answer should be: $$x(t)=\frac{1}{4}t^2-\frac{1}{8}+\frac{9}{8}\cos{2t}+\sin{2t}$$ Which is also verified ...
1
vote
1answer
40 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
0
votes
0answers
33 views

Why don't we check the exactness of differential equation with Inspection cases?

When solving the differential equations which are reducible to exact differential equations using Inspection cases for example: Solve: $2xy^2 + ye^xdx = e^xdy$ The integrating factor would $1/y^2$ ...
1
vote
0answers
13 views

Using the method of isoclines with logistic equation to create direction field

I am a little unsure on how to use the method of isoclines to model $\frac{dp}{dt} = 3p-2p^2$. As far as I know I need to set $3p-2p^2 = c$ where $c$ is the slope of the field on that line. When I set ...
0
votes
1answer
25 views

Lowering the order of a linear differential equation

Let $$L(x) \equiv x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}(t)x'+a_n(t)x=0.$$ and let the following solutions be given: $x_1,x_2,...,x_m(m<n)$- linear independent solutions. Let's find: $x_{m+1}, ...
1
vote
1answer
17 views

how to write a function in terms of Heaviside step function

I'm reading Paul Online Notes. There's an example of writing a function in terms of Heaviside step function as follows: $$ f(t) = \begin{cases} -4 &\text{if } t < 6, \\ 25 &\text{if } 6 \le ...
6
votes
1answer
316 views

Bernoulli Differential Equation of Second Order

How one can solve a Bernoulli differential equation of second order? i.e., solve the DE \begin{align} \frac{{d^2 y}}{{dx^2 }} + p\left( x \right)\frac{{dy}}{{dx}} + q \left( x \right)y = g\left( x ...