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
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0
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y
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1
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y
′
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a
2
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x
)
y
″
+
⋯
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a
n
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y
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{\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.
Homework Statement
Quote:
" PDE: ∂u/∂x + ∂u/∂y = 0
The general solution is u(x,y) = f(x-y) where f is an arbitrary function.
Alternatively, we can also say that the general solution is u(x,y) = g(y-x) where g is an arbitrary function. The two answers are equivalent since u(x,y) = g(y-x) =...
dr/d(theta) + r*sec(theta) = cos(theta)
apparently the solution is (sec(theta) + tan(theta))*r = theta - cos(theta) + c
but i have no idea how to get there.
i am using a technique from the book, but it yeilds an answer with lots of e's and an integral i can't solve.
this is my work...
Homework Statement
Claim:
For the partial differential equation ∂u/∂x + ∂u/∂y = 0, the general solution is u(x,y) = f(x-y)
Homework Equations
N/A
The Attempt at a Solution
I remember that the number of arbitrary constants in the general solution should be the same as the order of the...
Hello, this is the first time I post here, I'm really stumped and tried everything, even my TI-89 calculator won't give me something nice XD
Homework Statement
Find the general solution of
dy/dt= 1/(ty+t+y+1)
Homework Equations
No relevant equations.
The Attempt at a...
I was interrested in the general solutions to the wave equation depending on only one spatial coordinate.
For one linear coordinate, the general solution is:
a f(x-ct) + b g(x+ct)
For one radial spherical coordinate, the general solution is:
a f(r-ct)/r + b g(r+ct)/r
I thought that...
The flow defined by a differential equation/general solution
I'm trying to find the 'general solution'(perhaps more accurately the flow)
of the following ODE:
x' = x(1-x)
i.e. \lambda_{max}(t, \tau,\xi) and the domain where it's defined \Omega
note:
The definition of 'general...
Homework Statement
determine the general solution of the equation:
2sinx +cosecx -3 = 0
Homework Equations
The Attempt at a Solution
Dont know what to do need help please.
Homework Statement
Given: sec\theta=\sqrt{10} where 0< \theta <90
and \sqrt{10}sin(A-\theta)=sinA-3cosA
Determine the solution of
6cosA +3 = 2sinA
for A \in [-180; 180], rounded off to one decimal digit.
Homework Equations
The Attempt at a Solution
3=2sinA -...
Homework Statement
h(x)=cos(x+30) and g(x) = -2sinx
Determine the general solution without the use of a calculator, if
h(x)=g(x)
Homework Equations
trig identities, trig ratios, double angle formulae, reduction formulae.
The Attempt at a Solution
cos(x+30)=-2sinx...
Homework Statement
Find the general solution to the differential equation in implicit form.
http://www.texify.com/img/%5Clarge%5C%21%5Cfrac%7Bdy%7D%7Bdx%7D%3D%28cosx-sinx%29e%5E%7Bcosx%2Bsinx%2By%7D.gif
Homework Equations...
Homework Statement
The question asks me to find the general solution to (x^2-1)y' + 2y = (x+1)^2, and to determine the largest interval over which this general solution is defined. It's the latter which is tripping me up.
Homework Equations
I've already found the general solution as...
Homework Statement
Find the general solution of the following ODE:
x^{2}Y''(x) + xY'(x) - CY(x)=0
where C is a constant.
Homework Equations
The Attempt at a Solution
First I want to do this in the case where C=0; this gives:
x^{2}Y''(x) + xR'(x)=0
How do i solve this...
Homework Statement
Find the general solution of the differential equation,
y' + y = be^(-λx)
where b is a real number and λ is a positive constant.
Homework Equations
y' + P(x)y = Q(x)
Integrating factor: e^(∫P(x) dx)
The Attempt at a Solution
Let P(x) = 1, Q(x) =...
Homework Statement
Find the general solution of the first-order differential equation,
y' + 3y = 2xe^(-3x)
Homework Equations
y' + P(x)y = Q(x)
Integrating factor = e^(∫P(x) dx)
The Attempt at a Solution
Since it's already in the form y' + P(x)y = Q(x),
the integrating...
Homework Statement
Find the general solution of the differential equation specified.
\frac{dy}{dt} = ty.
The Attempt at a Solution
I already know the answer to be ke^{t^{2}/2}, but can't figure out how it got here. I'm rusty with my integrals and am just really starting diff eqs...
1. Homework Statement + relevant equations
I have to solve
http://home.vs.moe.edu.sg/linl/eqn1.gif
sigma, pinfinity, rho, N are constants. To make things easier for us, we are allowed to treat T as a constant.
2. The attempt at a solution
Treat T as constant...
Homework Statement
Find the general solution of (D^4 - I)^2(D^2 - 4D + 13I)^2(y) = 0
2. The attempt at a solution
My issue with this problem is that I have no clue as to what the I's mean. I have become familiar with D being used notationally with differential equations, but the...
1. The problem statement
Given the general solution:
y==(2*c*e^(2x)) / (1+c*e^(2x))
find the EDO.
2. The attempt at a solution
Im tried isolate c*e^(2x), using implicit differentiation:
y + y*c*e^(2x)==2*c*e^(2x)
y'+y'*c*e^2x + 2*y*c*e^(2x)==2*2*c*e^(2x)
y'==c*e^(2x) * (4-y'-2y)...
I was just wondering, if I had a matrix in reduced-row echelon form, say,
1 0 0 2
0 1 3 1
0 0 0 0
then I could write the general solution as a1= -2a4 , a2= -3a3-a4, with a3 and a4 defined in terms of these. (I obtained this solution by putting a1, a2, a3 and a4 under the first...
I have Abel equation of second type. I've found one particular solution for sure, maybe two. Is it possible to construct the general solution from those with integration or whatever?
I know that is possible for Ricatti ODE and since Ricatti is particular case of Abel, I was wondering if...
Homework Statement
Find the general solution to Laplace's equation in spherical coordinates, for the case where V depends on on r. Do the same for cylindrical coordinnates assuming V depends only on r.
Homework Equations
Laplace's Eq (spherical): 1/r^2 (d/dr)(r^2(dV/dr)) +...
Hi all.
Can the general solution of a linear ordinary differential equation be expressed in terms of its initial conditions?
It seems that I have seem this kind of representation.
It makes "some sense" to me but I hope to know if there is some "proof" or explanation of why it can be?
To be...
Homework Statement
Hi all.
I have the following equation containing the variables x and y, where A, c > 0 and B is a constant, which I am not told anything about:
-c^2x^2+y^2+2By+A=0.
I wish to find two solutions {x_1, y_1} and {x_2, y_2}, to this problem and they must be linearly...
starting with question of find the general sollution of the differential equation
X2y'=y2+3xy+X2 would an acceptable answer be y=-x(ln|x|+c+1) i would show all my working but my camera isn't working so i'll save you must the trouble and just skip to a part that i know is correct where...
Homework Statement
2xy" + y' + xy =0 xo=0
find the indicial equation, recurrence relation and series solution.
Homework Equations
The Attempt at a Solution
I've done most of the work but i don't know how to use a recurrence relation to obtain a y1 and y2 series solution...
y''+xy'+y=0
PS:
Is there a way to prove that one has found the "most" general solution by only using a particular method (e.g., integrating factors), with the possiblity of obtaining a different general solution using a different method?
For example, using integrating factor in the ODE...
Find the general solution of the ordinary differential equation.
y'' - 7y'+ 6y = 2e^(3t) + te^(t)
First i found GS(H) by lettings y = e^(cx)
and got GS(H) = Ae^(6t) + Be^(t)
i then found PS(IH) y'' - 7y'+ 6y = 2e^(3t) by letting y = ae^(3t)
and got PS(IH) = -(1/3)e^(3t)
Now my...
Im looking for a general solution to the following equation but i can't seem to get an answer.
3xdy = (ln(y^{6}) - 6lnx)ydx
I've got this far anyway..
\frac{3x}{y} dy = (ln(y^{6})-ln(x^{6})) dx
\frac{dy}{dx} = \frac{y}{3x} . 6ln\frac{y}{x}
\frac{dy}{dx} = \frac{y}{x} ...
What if I have the general solution and I want to find the differential equation where it came from?
Say for example my general solution is y=sin(ax + b) with a and b constants.
How could I find the differential equation?
Homework Statement
Given the matrix A=[...], find the general solution of; A[x;y;z]=[0;0;0]
Homework Equations
The Attempt at a Solution
This question has me lost. When I look at that it tells me x, y and z are all equal to 0. Is there something I'm missing, perhaps in the words...
im asked to find the general solution to the equation below and after find the implicit form I've done some work on it and just wanted to see if I am going in the right direction.
the equation
dy/dx = e^-3y cos x (1+sin x )^2
My attempt at a general solution
1/3 e^3 = (sin x...
Is there a general solution for gravity force in Alderson Disk system?
Given: Alderson Disk system, disk inner radius R1, outer radius R2, thickness T, density Rho, star mass M (irrelevant due to superposition?).
http://orbides.1gb.ru/orbf/ad1.jpg
How to find amount and direction of...
[b]1. The question asks me to show that e^x is a solution of xy'' - (2x+1)y' + (x+1)y=0 and find the general solution.
[b]2. I managed to simplify the equation to u''xe^(x) - u'e^(x) = 0 by letting y=ue^(x) and finding the differentials and substituting them in.
I've then let z=u...
Hi,
I'm having a bit of trouble with a problem here.
The question is: Use the Method of undetermined coefficients to Find the general solution to th system:
dx/dt = y + e^t
dy/dt = -2x + 3y + 1
I've got the homogenous solution fine, however I'm having a bit of difficulty with the...
Homework Statement
Find the general solution of the following differential equation:
x.(dy/dx) = y + sqrt.[(x^2) - (y^2)]
Homework Equations
I'm working through my excerise book and have been able to get through quite a few differential equations with success, but this one really does...
Homework Statement
Find the PDE for this general solution:
U(x,y) = Phi(x+y) + Psi(x-2y)
Homework Equations
The Attempt at a Solution
I let my xi = x+y and my eta = x-2y and found that both roots are {-1,1/2}. From that I multiplied: (dy/dx - root1)*(dy/dx - root2) to give me the...
Hello, I'm writing an application for a java class that solves the problem where you are given n jugs of arbitrary sizes and have to come up with the steps to reach a certain value.
I have figured out(read: did research) how to do this in a different way than the original, but it requires...
I have a question that involves a system of equations that I can't figure out
Give the general solution of the set of equations below:
x'=2x
y'=-x+3y
z'=2x-4y+6z
Hint: While you can use eigenvalues and eigenvectors for this one, there is an easier way to do it.
That's where I'm...
First it asks a few questions about what if it were a classical particle approaching the barrier. Much of this I understand and am OK with. Then we start treating the particle as a quantum thing so its governed by the TI Schrodinger EQ.
So, what it wants me to do which I am a bit unsure about...
Howdy,
I have been asked to find the general solution of the following matrix (pic attached).
The matrix does not have an inverse, so I am a bit confused guys. Cheers and thanks in advance!
Hi, can someone please help me with the following ODE? I need to find the general solution.
y = xy' + \frac{1}{{y'}}
Rearranging I get a quadratic in dy/dx.
x\left( {\frac{{dy}}{{dx}}} \right)^2 - y\left( {\frac{{dy}}{{dx}}} \right) + 1 = 0
\frac{{dy}}{{dx}} = \frac{{y \pm...
assuming that the circuit will lead to infinite, but converging, and R1,R3,R5 are in increasing magnitude, R2 is constant, is there a general formula for this problem?
Here is my question
Find the general solution to the differential equation
y''-18y'+80y=0
Express the solution in terms of the variable t. Give the smaller root first.
My problem is that I don't know what general solution mean?
what does it mean?
And where does the t come from?
1) I need to find the general solution, using the method of seperating the variables of the following diff Equ:
dy/dx = (2xcos x)/y where y>0
Is the answer = 2(cos x + xsinx)
2) If y = 2 when x = 0, find y in terms of x
Could someone help me on this one
3) Explain why your answer...
These are differential equations problems. They are now "redo's", so I have hints from the grader that I don't understand.
First problem:
(x^2 + 1)y'' + 6xy' + 4y = 0
After isolating C_n+2, I have this.
(n+2)(n+1)C_n+2 * X^n = -n(n-1)C_n * X^n - 6nC_n * X^n + 4C_n
C_n+2 = [-n(n-1)C_n - 6nC_n +...
hi all,
how do you find the particular integral of
\ddot{c} + \alpha c = \frac {\lambda L} {2lm} - g
I can find the complementary function of the above. Not sure what to do from here tho.
I have a 2nd order homogenous P.D.E:
(d^2)V/((dx)^2) + (d^2)V/((dy)^2) + 6(d^2)V/(dx dy) = 0
where all derivatives are partial derivatives. I need to transform this to form
a*(d^2)f/(dX^2) + b*(d^2)f/(dY^2) = 0 where a, b are constants and the derivatives are again partial, and f(X,Y) =...