What is General solution: Definition and 311 Discussions
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
a
0
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x
)
y
+
a
1
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x
)
y
′
+
a
2
(
x
)
y
″
+
⋯
+
a
n
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x
)
y
(
n
)
+
b
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x
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=
0
,
{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}+b(x)=0,}
where a0(x), …, an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, …, y(n) are the successive derivatives of an unknown function y of the variable x.
Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives.
A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any.
The solutions of linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.
TL;DR Summary: I have to find an equivalent resistance of the circuit below, dependent on the amount of ##R_3## - resistors.
Here is the circuit:
I think there is no general solution. When I want to calculate it, I have to do...
My take;
##ξ=-4x+6y## and ##η=6x+4y##
it follows that,
##52u_ξ +10u=e^{x+2y}##
for the homogenous part; we shall have the general solution;
$$u_h=e^{\frac{-5}{26} ξ} f{η }$$
now we note that
$$e^{x+2y}=e^{\frac{8ξ+η}{26}}$$
that is from solving the simultaneous equation;
##ξ=-4x+6y##...
For the 1 dimensional wave equation,
$$\frac{\partial^2 u}{\partial x ^2} - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$
##u## is of the form ##u(x \pm ct)##
For the 3 dimensional wave equation however,
$$\nabla ^2 u - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$It appears...
Hello guys I hope you all are doing well. :)
I found below question in a book by Martin Braun "Differential Equations and Their Applications An Introduction to Applied Mathematics (Fourth Edition)"
The question :
The Bernoulli differential equation is (dy/dt)+a(t)y=b(t)y^n. Multiplying through...
Hi , I'd like a little bit of clarification about Section 2.6 from Jackson's classic book on E & M.
Section 2.6 starts out with the problem of a "conducting sphere" near a point charge, but then it confusingly veers away to a problem where potential is prescribed to vary with azimuth and polar...
We know
$$
K(x,t) = \frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})
$$
is a solution to the heat equation:
$$
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}
$$
I would like to ask how to prove:
$$
u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)f(y)dy
$$
is also the solution to...
Find the general solution of
$y'−2y=t^{2}e^{2t}$
and use it to determine how solutions behave as
$t \to \infty$
ok presume the first thing to do is to find $u{x}$
$\exp{\displaystyle\int{2} y}=e^{-2} or \dfrac{1}{e^2}$
Hello everyone! I have two questions which had bothered me for quite some time. I am sorry if they are rather trivial.
The first is about the general solution of the hydrogen atom schrödinger-equation: We learned in our quantum mechanics class that the general solution of every quantum system...
Problem statement : Given the equation ##\sin\theta+2\cos\theta=1##, find the general solution for the angle ##\theta##.
Attempt : For the general case where we have ##a\sin\theta+b\cos\theta=c##, the line of approach is to take ##a=r\cos\alpha## and ##b=r\sin\alpha## wherein we will have...
$\tiny{307w.3.2.5.3}$
https://mathhelpboards.com/{http%3A//faculty.sfasu.edu/judsontw/ode/html-snapshot/linear02.html
Find the general solution of each of the linear system
\begin{align*}
x' & = -3 x + 4y\\
y' & = 3x - 2y
\end{align*}
$A=\begin{pmatrix}-3&4\\ 3&-2\end{pmatrix}...
Hi,
I was trying to do the following problem.
My attempt.
Finding the reduced row echelon form for the system above.
I do not see any way to proceed any further. The following is the solution presented in solution manual. How do I proceed to get the following answer?
$\tiny{27.1}$
623
Find a general solution to the system of differential equations
$\begin{array}{llrr}\displaystyle
\textit{given}
&y'_1=\ \ y_1+2y_2\\
&y'_2=3y_1+2y_2\\
\textit{solving }
&A=\begin{pmatrix}1 &2\\3 &2\end{pmatrix}\\...
This equation, is non linear, non-separable, and weird. I would like to have a direction to start working on this.
I tried writing sin(2y) = 2sin(y)*cos(y).
See,
##xy' = x^3sin^2(y)-2sin(y)cos(y)##
Can't separate.
Writing in this way:
##(x^3sin^2y-sin2y)dx-xdy=0##
Also, I checked that it is...
Find the general solution of the given differential equation, and use it to determine how solutions behave as $ t\to\infty$ $y'+y=5\sin{2t}$
ok I did this first
$u(t)y'+u(t)y=u(i)5\sin{2t}$
then
$\frac{1}{5}u(t)y'+\frac{1}{5}u(t)y=u(i)\sin{2t}$
so far ... couldn't find an esample to follow...
Since the spherical wave equation is linear, the general solution is a summation of all normal modes.
To find the particular solution for a given value of i, we can try using the method of separation of variables.
$$ ψ(r,t)=R(r)T(t)ψ(r,t)=R(r)T(t) $$
Plug this separable solution into the...
For a harmonic oscillator with a restoring force with F= -mω2x, I get that the solution for the x-component happens at x=exp(±iωt). But why is it that you can generalise the solution to x= Ccosωt+Dsin(ωt)? Where does the sine term come from because when I use Euler's formula, the only real part...
Find a general solution to the system of differential equations
\begin{align*}\displaystyle
y'_1&=2y_1+3y_2+5x\\
y'_2&=y_1+4y_2+10
\end{align*}
rewrite as
$$Y'=\left[\begin{array}{c}2 & 3 \\ 1 & 4 \end{array}\right]Y
+\left[\begin{array}{c}5x\\...
So in my previous math class I spotted on my book an exercise that I couldn't solve. We had to find the general solution for the differential equation. This was the exercise: 4y'' - 4y' + y = ex/2√(1-x2)
Can anyone tell me how to solve this step by step?
$\tiny{28.1}$
2000
Find the general solution to the system of differential equations
\begin{align*}\displaystyle
y'_1&=y_1+5y_2\\
y'_2&=-2y_1+-y_2
\end{align*}
why is there a $+-y_2$ in the given
ok going to take this a step at a time...
Hello,
I'm trying to find the general solution of this homog. system w/ constant coefficients. I can't even get past the first step, which is to find the eigenvalues. As far as I know, there are a few approaches:
1) solve det(A-λI) = 0
2) solve the trace determinant plane equation (which is...
Find the general solution to the system of differential equations
$\begin{cases}
y'_1&=2y_1+y_2-y_3 \\
y'_2&=3y_2+y_3\\
y'_3&=3y_3
\end{cases}$
let
$y(t)=\begin{bmatrix}{y_1(t)\\y_2(t)\\y_3(t)}\end{bmatrix}
,\quad A=\begin{bmatrix}
2 & 1 & -1 \\
0 & 3 & 1 \\...
Has anyone formulated a general solution to the time-independent Schrödinger equation in terms of the potential function V(r), and if so, what is it?
For any type of V(r).
So, instead of a differential equation, a direct relationship between the wavefunction and the potential.
$\tiny{2.1.9}$
2000
Find the general solution of the given differential equation, and use it to determine how
solutions behave as $t\to\infty$.
$2y'+y=3t$
divide by 2
$y'+\frac{1}{2}y=\frac{3}{2}t$
find integrating factor,
$\displaystyle\exp\left(\int \frac{1}{2}...
I've found the general solution to be y(x) = C1cos(x) + C2sin(x).
I've also found a recursion relation for the equation to be:
An+2 = -An / (n+2)(n+1)
I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this?
Any help would be...
$\tiny{3.1.1}$
find the general solution of the second order differential equation.
$$y''+2y'-3y=0$$
assume that $y = e^{rt}$ then,
$$r^2+2r-3=0\implies (r+3)(r-1)=0$$
new stuff... so far..
Could someone please provide guidance on how to begin this problem? I've attached the preface to the assignment question.Show that the general solution of the differential equation
y″(x)=−y(x)
is
y(x)=Acos(x)+Bsin(x)
where A and B are arbitrary constants. Hint: You'll need the Taylor series...
Homework Statement
Homework Equations
General Formula for Tan(a)=Tan(b)
The Attempt at a Solution
See the question I have uploaded.
I have tried solving it this way,
Firstly I applied the Quadratic Formula to get,
Now we have two cases,
CASE-1
When
So General Formula here will...
Dear all
I've been trying to work out the general solution to a 2nd order ODE of the form
f''(x)+p(x)*f(x)=0
p(x) is a polynomial for my case. I believe series method should work, but for some reason I would prefer a general solution using other methods. I'll be much appreciative for any help...
Homework Statement
The question I am working on is number 3 in the attached file. There are two initial conditions given: at time = 0, x(t) = D and x'(t) = v 'in the direction towards the equilibrium position'. Does that last statement mean that when I substitute the second IC in, I should...
Hi!
I'm having a hard time with general solutions of a certain type, maybe it has to do with the constant in the particular solution?
The equation is
y''+2y'+y=xe-x+1
Is it right to assume that yh=(C1x+C2)e-x ?
For the particular solution, I've tried all kinds of guesses for the form:
yp=...
Homework Statement
Find the general solution of the given differential equation. Give the largest interval I over which the interval is defined.Determine wether there are any transient terms in the general solution
5.
\frac {dy}{dx} + 3x^2y = x^2
Homework EquationsThe Attempt at a Solution...
Homework Statement
Killing Equation is: ##\nabla_u K_v + \nabla_v K_u =0 ##
In flat s-t this reduces to:
##\partial_u K_v + \partial_v K_u =0 ##
With a general solution of the form:
##K_u= a_u + b_{uc} K^c ##
where ##a_u## and ##b_{uv}## are a constant vector and a constant tensor...
Homework Statement
Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions ## g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta} ##
Using the method of Green's functions, and ## S(\theta,t)= \frac{1}{\sqrt{4\pi...
Homework Statement
Find the solution of the inequality ## \sqrt{5-2sin(x)}\geq6sin(x)-1 ##
Answer: ## [\frac{\pi(12n-7)}{6} ,\frac{\pi(12n+1)}{6}]~~; n \in Z##Homework Equations
None.
The Attempt at a Solution
There are two cases possible;
Case-1: ##6sin(x)-1\geq0##
or...
$\tiny{17.1.08}$
$\textrm{ Find the general solution of the given equation?}$
\begin{align*}\displaystyle
y''+100y&=0
\end{align*}
$\textit{The auxiliary equation is:}$
\begin{align*}\displaystyle
x^2+100x&=0\\
x(x+100)&=0\\
x&=-100
\end{align*}
$\textit{Answer by EMH}$...
Homework Statement :[/B]
Find the general solution of the Trigonometric equation: $$3\sin ^2 {\theta} + 7\cos ^2 {\theta} =6$$
Given andwer: ##n\pi \pm \frac {\pi}{6}##
Homework Equations :[/B]
These equations may help:
The Attempt at a Solution :[/B]
Please see the pic below:
It...
Homework Statement :[/B]
Find the general solution of the equation: $$\tan {x}+\tan {2x}+\tan {3x}=0$$
Answer given: ##x=## ##\frac {n\pi}{3}##, ##n\pi \pm \alpha## where ##\tan {\alpha} = \frac {1}{\sqrt {2}}##.
Homework Equations :[/B]
These equations may be used:
The Attempt at a...
Homework Statement :[/B]
Find the general solution of the Trigonometric equation $$\sin {3x}+\sin {x}=\cos {6x}+\cos {4x} $$
Answers given are: ##(2n+1)\frac {\pi}{2}##, ##(4n+1)\frac {\pi}{14}## and ##(4n-1)\frac {\pi}{6}##.
Homework Equations :[/B]
Equations that may be used:
The...
Hello,
If I have an x^2 graph that goes from 0 to a point a. Is there a general solution to where the area of the left side is equal to the area of the right?
Homework Statement
"By choosing the lower limit of integration in Eq. (28) in the text as the initial point ##t_0##, show that ##Y(t)## becomes
##Y(t)=\int_{t_0}^t(\frac{y_1(s)y_2(t)-y_t(t)y_2(s)}{y_1(s)y_2'(s)-y_1'(s)y_2(s)})g(s)ds##
Show that ##Y(t)## is a solution of the initial value...
Homework Statement
An equation of motion for a pendulum:
(-g/L)sinΦ = Φ(double dot)
Homework Equations
L = length
g = gravity
ω = angular velocity
Φο = initial Φ
The Attempt at a Solution
The solution is Φ=Asinωt+Bcosωt
solving for A and B by setting Φ and Φ(dot) equal to zero respectively...
Homework Statement
So they want me to obtain the general solution for this ODE.
Homework Equations
I have managed to turn it into d^2y/dx^2=(y/x)^2.
The Attempt at a Solution
My question is, can I simply make d^2y/dx^2 into (dy/dx)^2, cancel the power of 2 from both sides of the equation...