Questions tagged [ordinary-differential-equations]
For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.
42,179
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How I can solve this nonlinear second order ODE?
I am trying to solve a non-linear ODE. Let $g(x)$ is an arbitrary smooth real function, I want to solve the following equation
$2f(x)f''(x) +4(f'(x))^2=g(x)$
for $f(x)$.
Multiplying $\dfrac32 f(x)$, ...
- 11
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1
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40
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Simple ODE question. A student learns 150 Chinese characters each month and forgets 10% of what they have learned
A student learns 150 Chinese characters each month and forgets 10% of what they have learned each month as well.
Construct a differential equation suitable from the problem
Find a solution with a ...
- 975
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$(\beta_n)_{n \geq 0}$ converges uniformly to the solution of $x' = F(t,x)$, variation of Picard iteration?
This is exercise 2.7. from Differential Equations: A Dynamical Systems Approach to Theory and Practice by Marcelo Viana and José Espinar.
Let $F \colon \mathcal{U} \to \mathbb{R}^n$ be continuous and ...
- 1,066
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votes
1
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58
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Is there a way to show that this ode system is asymptotically stable?
Suppose we have
$$\dot{x} = -\frac{x}{y+a} $$
$$\dot{y} = -y$$ for $a>0$,
Is the above system asymptotically stable?
Now, I know that we can solve for $y$ as
$$y = y(0) e^{-t}$$
and we can choose ...
- 57
0
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1
answer
46
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Show that if $X$ is a solution of $X'=A(t)X$ with $A$ a matrix and $X(0)$ with positive coefficients, $X(t)$ has positive positive coefficients
Let $A : \mathbb R^+ \to M_n(\mathbb R^+) \in C(\mathbb R^+,M_n(\mathbb R^+) )$ and $X$ a solution of $X'=AX$ such that $X(0)$ with positive coefficients $(\ge 0)$.
I need to show that : $\forall t\in ...
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1
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43
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What is the general solution to the homogeneous differential equation $2y''+2y'+3y=0\ $? [closed]
The general solution I came up with for $2y''+2y'+3y=0$ is $$e^{-x/2}[c_1\cos(x\sqrt{5}/2)+c_2\sin(x\sqrt{5}/2)]$$where roots $r$ are $ -1/2 + i\sqrt{5}/2$ and $-1/2 - i\sqrt{5}/2.$
Am I correct?
- 31
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1
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How to prove exponential functional identity knowing that it is a solution to a first order ODE and knowing its Taylor expansion
Establish the identity
$$E(ax)E(bx) = E[(a+b)x]$$
knowing that
$y = E(px)$ satisfies $y' - py = 0$ and
$E(px) = \sum_{n=0}^\infty\frac{(px)^n}{n!}$
An additional hint the textbook gives ; "...
- 1
1
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1
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Confusion about solving linear second order ode with boundary condition.
So I have the following problem in a textbook I have to solve. Sadly I'm not sure what to do.
Given is $y^"(x)=\lambda y(x)$ with $\lambda \in \mathbb{R}^+$. I'm supposed to find a solutions $(y(x), \...
- 33
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Can generating functions be used to solve evolution matrix differential equations and recurrence relations of matrices?
Generating functions seem to be a powerful tool in discrete mathematics for solving differential equations and recurrence relations. I've been trying to figure out if these methods can be expanded to ...
- 31
4
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41
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almost-Newton flows are Newton flows where the chain-rule is 'forgotten', yet its solutions are roots of f anyway, when does this work?
The differential equation for the Newton flow $z (t)$ of $f (t)$ is given by
\begin{equation}
\dot{z} (t) = - \frac{f (z (t))}{\frac{d}{d t} f (z (t))} = -
\frac{f (z (t))}{\dot{f} (z (t)) \dot{z} ...
- 127
3
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0
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57
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Stability of numerical solution of ODE
I want to solve the ODE
\begin{array}{ll}
-u''(x)=f(x) & x\in (0,1) \\
u(0)=g(0) \\
u(1)=g(1)\, \\
\end{array}
with finite differences using $u''(x)\approx \frac{u(x-h)-2u(x)+u(x+h)}{h^2}$.
To ...
- 484
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votes
1
answer
81
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Using the power series method to solve $y''-(1+x)y'-y=0$
I am trying to solve the following second order differential equation.
$$y''-(1+x)y'-y=0$$
I believe this differential equation can be solved using the power series method. So, suppose $y(x)=\sum_{k\...
- 89
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0
answers
21
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SDE in 2-dimensional time domain
Usually an SDE $dX_t = f(X_t,t) + g(X_t,t)dW_t$ is formulated in 1-dimensional time domain, $t\in \mathbb{R}$. However, in principle, the time could be also a subset of $\mathbb{R}^2$, $t=(t_1,t_2) \...
- 63
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2
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36
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Integrating w.r.t. two different variables - a physics problem on the maximum height of a vertically thrown object
I'm trying to derive an equation for the height an object reaches after being vertically thrown upwards, where the two forces acting on it are gravity and quadratic air resistance, so that
$$
ma=-mg-...
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0
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25
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First order Sturm-Liouville eigenvalues and eigen functions [closed]
Solve:
$$y'+6\lambda x^5y=0$$
$$y(0)+6y(3)=0$$
Does anyone know how to calculate the eigenvalues and eigenfunctions of this particular Sturm Liouville problem?
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Calculating the maximum height a person can jump on the moon, given the maximum height they jumped on earth [closed]
A person on earth performs a high jump of 2.41m. Assuming the person is wearing an ideal spacesuit that does not adversely affect athletic performance, the acceleration due to gravity on the moon is 1....
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31
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A differential equation with infinitely many characteristic roots
Let us consider the following neutral differential equation :
$$ \dot{x}(t)-\dot{x}(t-\tau)=ax(t)+bx(t-\tau)$$
where $\tau >0$ is a constant delay, and $a,b\in \mathbb{R}$.
It is a well-known fact ...
- 172
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28
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Differential equation - Laplace with Dirichlet condition on a rectangle
In a rectangular domain $R$ with sides $a$ and $b$, with $b^2/a^2$ irrational, we look at the differential equation $- \Delta u = \lambda u$ with Dirichlet boundary conditions. How to show that the ...
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1
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83
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Inequality of $|\sin(a+b)-\sin(c+d)|<|(a+b)-(c+d)|$
I am reading a solution with this inequality:
$$|\sin(a+b)-\sin(c+d)|\le|(a+b)-(c+d)|$$
The solution just says this holds, but I don't quite understand how?
I am also trying to know if it can be ...
- 75
1
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0
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31
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General Solution to Secular Terms in ODE System: Inductive Proof
Attached is a proof of the general solution to a system of differential equations that has secular terms as a result of repeated eigenvalues, and hence solved using a Jordan Normal form.
I can follow ...
- 364
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1
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28
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Reduction of order of a second order linear ODE
Let there be a second order linear homogeneous ordinary differential equation in its general form,
$y{''}+p(x)y{'}+q(x)y=0$ When the equation is of a particular form $F(x,y{'},y{''})=0$, I can easily ...
- 133
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28
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How can I find out if a curve is orthogonal using differential equations
How can I find out that the curves included in the $y=c(2x+c)$ are orthogonal when $c$ is a random constant?
I've tried defining $c$ with $x$ and $y$ but I failed.
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2
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66
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How to integrate $xy' = \sqrt{x^2-y^2}+y$
I have to integrate:
$xy' = \sqrt{x^2-y^2}+y$
It is supposed to be an homogeneous differential equation, but I can't see the variable change to make it happens. I tried:
$y' = \sqrt{1-(y/x)^2} + y/x$
...
4
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1
answer
53
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Variational formulation, weak formulation
I'd like to find the weak formulation of the problem
$-u''+au=f$ on $(0,1)$
$u(0)=0$
$u'(1)=b$
$a>0$
and show that there exists a unique solution using Lax-Milgram.
What I did:
By multiplying ...
- 199
2
votes
1
answer
46
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Periodic orbit in $C^1$ vector field on annulus
The following problem has been taken from Differential Equations, Dynamical Systems, and an Introduction to Chaos (Hirsch, Devaney, Smale).
Let $A$ be an annular region in $\mathbb{R}^2$. Let $F$ be ...
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3
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How do I solve this differential equation? $y'' - 2y' +4y = e^x\sin (x)$
I have calculated the wronskian and while finding particular solution using variation of parameters. The integral becomes $$\int \sin(\sqrt{3}x)\sin(x)dx$$ I am stuck here. Have I done anything ...
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0
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How to solve $x+yy'+y'y''=0$ [closed]
Solve $x+yy'+y'y''=0$, where $y'$ and $y''$ are first and second derivatives of $y$ wrt $x$
I have only studied high school differential equations which deals mostly with degree 1 and order 1 ...
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Who could help me solve this differential equation? [closed]
How do I solve this differential equation?
$$\dfrac{y''}{y}= -2 \left( \dfrac{y'}{y} \right)^2$$
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Is the fact that $e^{ix}$ and $\cos(x) + i \sin(x)$ have the same derivative and a point in common enough to imply they’re equal
If we let
$$f(x) = e^{ix}$$
and
$$g(x) = \cos(x) + i \sin(x),$$
then
\begin{align*}
f(0) &= g(0) = 1,\\
f’(x) &= if(x),\\
g’(x) &= ig(x).
\end{align*}
Is this enough to imply they’re the ...
1
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1
answer
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Solving non-linear ordinary differential equation
I am attempting to solve the differential equation:
$$(y')^2 = 1 $$ with the initial condition $y(0)=0$
By moving into the Laplace domain we get:
\begin{align*}s^{2}Y(s)^{2} &= \frac{1}{s}\\Y(s)^{...
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answers
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Generating Function for the number of $n$-permutations whose square is the identity permutation.
I am learning the concept of generating functions and am working on the following problem:
Let $r(n)$ be the number of $n$-permutations whose square is the identity permutation. We proved that
$$r(n+...
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2
votes
1
answer
28
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Reducing a system of ordinary differential equations by imposing relations between the variables
Let's assume that we have the following system of O.D.E.
\begin{align}
x' = f_1(x,y,z,v),\ y' = f_2(x,y,z,v),\ z' = f_3(x,y,z,v),\ v' = f_4(x,y,z,v)
\end{align}
for suitable smooth functions $f_1$,...
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Equation with variable within an integral sign
This seems as a very easy equation but I am having a hard time solving it. I need to solve for $\beta(t)$ . $q_i(t)$'s are given. (Extra fact : $\int q_i(t)^2dt=L_i$ where $L_i$'s are constants)
Have ...
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Defining formulas for first-order linear differential equations.
When defining the formulas for the first-order linear differentiable functions we are necessitated to define a equation that satisfies $u'(x)$ = $u(x)p(x)$ so then the product rule can be applied. And ...
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1
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Integrating $y' = \sin^2 y$
I'm trying to solve: $$y' = \sin^2 y$$
It is a separable variable differential equation, so I arrive to
$$\int\frac{dy}{\sin^2 y}=\int dx$$
I use the identity $$\sin^2 y=\frac{1-\cos(2y)}{1}$$ and ...
0
votes
2
answers
137
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How to solve $dx/dt=x^3-x$
I want to solve the differential equation:
$$
\frac{dx}{dt}=x^3-x.
$$
If we seek a solution $x$ such that there exist $t_0\in\mathbb R$ verifying $x(t_0)$ equal to $0$, $1$ or $-1$, we find using the ...
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0
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In the definition for sensitivity to initial conditions what exactly does the distance between trajectories mean?
I've seen this definition for sensitive dependance in Modeling Life (Garfinkel et al, 2010):
$$d(M_{t} - N_{t}) = e^{\lambda * t} * d(M_0 - N_0)$$
or alternatively from Wikipedia:
$$ {|\delta \mathbf {...
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Estimates of Matrix Differential equation
I’m working with the following matrix differential equation:
A’’ + A / |A| = 0, A is three-dimensional square matrix and |A|>0.
According to Picard theorem, there is a unique solution. But I want ...
- 1
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0
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ODE particular solution confusion
I've been having trouble with making correct guesses on particular solutions. More specifically on this question: $y''-4y'+4y=12e^{2t}$ , which is example one on this page https://math.jhu.edu/~brown/...
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2 by 2 coupled ODEs with non-constant coefficients
I have the following coupled linear DE that I wish to solve. However, their coefficients are non-constant, namely, I wish to solve
$$
\frac{d}{dx}\begin{bmatrix}y(x) \newline z(x)\end{bmatrix} = \...
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2
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65
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How to solve $\frac{d^2y}{dx^2} = (x^2 - k_0)y$
What method can be used to solve and plot the y over x graph satisfying:
$$\frac{d^2y}{dx^2} = (x^2 - k_0)y$$
for a given constant $k_0$?
In particular, I am trying to reproduce the following:
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0
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25
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Equilibrium stability
Can somebody help me to prove that the equilibrium is stable?
The ODE:
$dC/dt = \lambda- kVC - \delta C$
$dC_p/dt = kVC - v C_p - \delta C_p$
$dV/dt = x C_p - kVC - zV$.
The equilibriim is:
$C = \frac ...
- 1
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vote
0
answers
20
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Second Order Homogenous ODE solve with Real Analysis Integration topics [duplicate]
Question:
Suppose that $u\in C([a,b])$ is twice continuously differentiable, $V\in C([a,b])$, $V(x)\geq0$ for all $x \in [a,b]$ and
$$
-u''(x) + V(x)u(x)=0, \;\; x\in [a,b],
$$
$$
u(a)=u(b)=0
$$
Prove ...
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votes
1
answer
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What exactly should the solution to $y^\prime = (1 - 2x)y^2, \ y(0) = -\frac{1}{12}$ be?
(Hey everyone, I'm new to Stack Exchange and this is actually my first question, so if I'm doing anything inappropriate regarding asking questions, please point that out for me.)
So I've been taking ...
- 53
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votes
1
answer
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+100
Green function jump conditions for second order differential equation
I been trying to find the Green's function for a particular problem.
For the equation
$$
q(x) \frac{\mathrm{d}^2u(x)}{\mathrm{d}x^2} + p(x) u(x) =0 \tag{1}
$$
where $q(x)$ and $p(x)$ are some ...
- 139
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votes
0
answers
33
views
Existence and uniqueness theorem applying the Runge-Kutta formula
I want prove that the problem
$$y'=f(x,y)$$
$$y(x_0)=y_0$$
It has a unique solution if $f$ is continuous in $[a,b] \times \mathbb{R}^n$ and it satisfies the Lipschitz condition with respect to the ...
3
votes
0
answers
63
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Find the Green’s function $G(x, t)$ using a Laplace transform in $t$
Given the wave equation: $G_{tt} - G_{xx} = \delta(x-x_{0})\delta(t-t_{0})$,
with conditions: $-\infty < x < \infty,t>0$,$G(x,0) = 0, G_{t}(x,0) = 0$. Use Laplace transform to find the Green'...
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votes
0
answers
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Prove that the solution to this differential equation must become constant once it hits 1.
I have the following conjecture which I hope to prove.
Conjecture. Let $f :[a,b]\to \mathbb R$ be differentiable on $[a,b]$ that satisfies
$$
f'(x) = {\left(1-f(x)^3\right)}^{\frac 1 3}
$$
for all $x\...
- 159
2
votes
2
answers
73
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Kernel and Differential Operator
I hope your day is great so far.
I have a question. I am given a Cauchy-Euler ODE
$$\big((x+2)^3\mathbf{D}^3+4(x+2)^2\mathbf{D}^2+3(x+2)\mathbf{D}+1\big)y=\frac{1}{x+2},$$
where $\mathbf{D}^n=d^n/dx^n$...
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2
votes
0
answers
61
views
Solving $\frac{d\mathbf{y}}{dt} + \mathbf{G}(t) \mathbf{y} = \mathbf{z}(t)$
Is there a known solution for the linear equation below?
$$
\frac{d\mathbf{y}}{dt} + \mathbf{G}(t) \mathbf{y} = \mathbf{z}(t)
$$
The variables $\mathbf{y}$ is a vector of $n$ elements, $\mathbf{z}$ is ...
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