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
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y
+
a
1
(
x
)
y
′
+
a
2
(
x
)
y
″
+
⋯
+
a
n
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x
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y
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n
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+
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.
Homework Statement I need to find the solution to x'' + cx' = f(t), for a general f.
Homework Equations
The Attempt at a Solution
Obviously first I solve the homogeneous part to give me A + B*exp(-ct). I also know that the particular solution is written as (1/c)int((1-exp(c(s-t))f(s))ds...
yIII+yII-yI-y = 0
I used the characteristic equation and got:
r3+r2-r = 0
r (r2+r-1) = 0
Which means that r = 0 is one root,
And the other factors from the polynomial are (-1-Sqrt(5))/2 and (-1+Sqrt(5))/2
This means that the final answer would be:
y = C1 Exp(0x) + C2 Exp((-1-Sqrt(5))/2) +...
Homework Statement
Let |\psi\rangle and |\psi '\rangle be solutions to the same Schrodinger equation. Show than, that c|\psi\rangle+c'|\psi '\rangle is the solution, where c and c' are arbitrary complex coefficients, for which holds: |c|^2+|c'|^2=1
The Attempt at a Solution
Now this follows...
Homework Statement
http://img811.imageshack.us/img811/1989/problem1.png
Homework Equations
All shown in the above link, AFAIK
The Attempt at a Solution
Not worried about part a.
For part b, when they say "assume the string is initially at rest" I took that to mean...
When I realize that I am going to have a singular matrix (after exhausting row swap options and maybe even some elimination steps) what about the matrix tells me whether or not I can have a general solution?
Homework Statement
Find the general solution of:
a) sin x = 1/\sqrt{2}
b) cos x = 0.5
The Attempt at a Solution
a) x = asin(1/\sqrt{2})
x = (π/4) + 2nπ
b) x = acos(0.5)
x = π/3 + 2nπ
Basically, my strategy was to solve for the basic angle, and then add multiples of the...
I need to solve d/dx(e^(-(x^2))*(du/dx))=u+x*(du/dx)
I get e^(-(x^2))*u''-2xe^(-(x^2))*u'=u+x*u'
u''-2x*u'=e^(x^2)*u+xe^(x^2)*u'
u''+(2x-xe^(x^2))*u'-e^(x^2)*u=0
but I have no idea how to approach the problem from here. Can somebody please help?
Homework Statement
I want to find the general solution of these two equations,
\ddot{y}=\omega\dot{z}
\ddot{z}=\omega\left(\frac{\mathbf{E}}{\mathbf{B}} - \dot{y}\right)
Homework Equations
These two equations are the result of quantitatively solving to find the trajectory of a charged...
Homework Statement
getting gen sol of
xy3zx+x2z2zy=y3z
solve cauchy problem
x=y=t, z=1/t
The Attempt at a Solution
i got gen sol F(C1,C2)=0 as
C1=x/z, C2=y4-x2z2
i inserted t for x and y and 1/t for z and ended up with
C1-2=1/(C22)
I'm unsure what to do from...
Homework Statement
y'''-3y'+2y=0
initial conditions y(0)=0, y'(0)=1,y''(0)=1
Homework Equations
Assume y=e^{rt}
The Attempt at a Solution
By the substitution I'm left with
r^3-3r+2=0
which gives me the roots of -2 and 1.
my question is a lot of times with this type...
Homework Statement
Solve y''+4y'+5y=0
find solutions for y(0)=1 and y'(0)=0
Homework Equations
Quadratic equation
The Attempt at a Solution
Hows this look ?
assume solution is in the form of y=ce^{rx}
substitute y=ce^{rx} into the equation...
Im having trouble with this question. can anyone explain please?
Homework Statement
y'' + 6y' + 9y = x*exp(-3x)3x
Homework Equations
Find the general solution.
Homework Statement
y' = \frac{y+y^2}{x+x^2}
Homework Equations
separation of variables
The Attempt at a Solution
I start with
y' = \frac{y+y^2}{x+x^2}
which is
\frac{dy}{dx} = \frac{y+y^2}{x+x^2}
next step is
dy = \frac{y+y^2}{x+x^2}dx
than I divide both sides by...
Homework Statement
(x^2)yy' = e^x
Homework Equations
general solution to the DE
The Attempt at a Solution
first i changed y' to dy/dx
(x^2)y(dy/dx) = e^x
then divided both members by x^2 and multiplied both members by dx
ydy = (e^x)dx/(x^2)
or
ydy = (x^-2)(e^x)dx...
I have this question, but don't know how to even start.
Suppose (M) is a linear system of 2 equations and 3 unknowns, where (2,-3,1) its solution.
Suppose (O) is a matching homogeneous linear system, where (-1,1,1) and (1,0,1) its solutions.
How can I find the general solution of (M)?
I'm...
Homework Statement
y''+4y'+4y= t+exp(-2t)
find the general solution for the differential equation
Homework Equations
The Attempt at a Solution
general solution is sum of complementary function and particular integral
frist finding complementary function
y''+4y'+4y=0...
Homework Statement
Find a general solution for y''' - 6y'' + 9y' = 0
Homework Equations
The Attempt at a Solution
I know that the general solution for a homogeneous DEQ is
Y(x) = c1y1(x) + c2y2(x) ... cnyn(x)
however, I am not given y1, y2 , or y3 so I am to assume that the...
Homework Statement
This is really a maths problem I'm having.
I need to get the general solution for the infinite square well in the form:
u = A cos(kx) + B sin(kx)
I found the general solution to be:
u = A exp(ikx) + B exp(-ikx)
Using Euler's formula:
exp(ikx) =...
y''' + y' = tan(t) 0<t<pi
I got yh = c1 + c2cos(t) + c3sin(t)
I'm trying to solve by Undetermined Coefficient method and Variation of Parameters method, but it didn't work
Homework Statement
Find the general solution
y'' + 4y' +4y = 5xe^(-2x)
The Attempt at a Solution
I got (5/2)x^3*e^(-2x) as a particular solution. But I checked online at wolfram alpha and it says the particular solution is (5/6)x^3*e^(-2x). Using method of undetermined coefficients.
Question: find the general solution of xX' = aX
i know it's kinda simple ode.. but, i just don't know y i can't get the correct answer..
solution..
xX' = aX
x dX/dx = aX
d/dx X = aX/x
X = ∫aX/x dx
X = aX ln |x| + C
the general solution is Ax^a
problem : i can't get the...
Homework Statement
"find the general solution of the equation: y'' + 3y' + 2y = 0
characteristic is:
r^2 + 3r + 2 = 0
solve quadratic:
(r+2)(r+1)
r = -2
r= -1
therefore GS of equation is: y = c_1e^-2x + c_2e^-x
thanks for any help
Homework Equations
The Attempt at a Solution
ok well I'm pretty much home and dry in this problem
the aim of this problem is to get the general solution for the ode below..
2u'' - xu' + u = 0 = g(x)
i started to solve it by rearranging the equation..
2u'' + u = xu'
apply Fourier transform..
2F(u'') + u^ = g^
(-2k^2)u^ + u^...
Homework Statement
1) Find the general solution to:
(t2D2 - 2tD - 28I)[y] = (-17 + 48t - 97t2 + 6t3)e-t
2) Find the general solution to:
(D + tI)2[y] = 3 + 3t + 6t2 + t3+ t4
Homework Equations
The Attempt at a Solution
1. I think for this one, I just need to distribute...
[b] Find the general solution of the following ODE:
dx/dt = 3x^(2) cos t
[b] Make x the subject of the solution.
[b] Heres my solution, is this correct?
dx/dt = 3x^(2) cos t
dx/3x^(2) = cos t dt
Integrating both sides gives:
ln (3x^(2)) = sin t + C
3x^(2) =...
Find y(t) assuming that y(0) = 4000
Homework Equations
This is what I know!
v = 20√10 x ((1+Ae^(t/√10))÷(1-Ae^(t/√10)))
and
A = -1 when finding particular solution to satisfy initial condition v(0).
Terminal velocity = 63.246 m.s^-2
The Attempt at a Solution...
[b]1. Using separation of variables show that:
v' = (cd/M)(v^2) - g has a general solution of:
v = 20SQRT10 x ((1+Ae^(t/SQRT10)/(1-Ae^(t/SQRT10))
Homework Equations
The Attempt at a Solution
Have attempted numerous times with little success help appreciated!
If we assume air resistance is negligible, the only force acting on a body is -Mg where g is the acceleration due to gravity ( negative because acting downwards). F = Ma becomes :
-Mg = M y''
which implies
y[B]''=-g
Question asks find the general solution for y.
Homework Equations...
At+1=(At+r)/(At+r+1)
A1=constant
I know I can set At+1=At=A and solve for a special solution.
What would be a general solution?
I am not taking a course in Difference Equation, and this is not my homework but I encounter a similar question and I reduce it to this form.
Thanks
Homework Statement
Show that:
\theta (\xi) = 1-\frac{1}{6} \xi^2 + \frac{n}{120} \xi^4 +...
is a general solution to the Lane-Emden equation. (assuming that the above equation converges)
Homework Equations
Lane-Emden equation:
\frac{1}{\xi^2}\frac{d}{d \xi} \left ( \xi^2...
Homework Statement
Find if it is true that the general solution to : y'' - y' = 0, where y(x),
can be written as : y(x) = c1 cosh(x) + c2 sinh(x), where c1 and c2 are real
arbitrary constants.
Homework Equations
differential equation solving
The Attempt at a Solution
I just...
This should be very simple, but I can find a simple example that violates my general conclusion. I clearly must be doing something wrong with my integration by parts. Any suggestions would be great.
Define a distribution such that the density;
\eta(\vec{x})=\int d\vec{k} f(\vec{x},\vec{k})...
Homework Statement
Find the general solution to the system:
ax+ by= 1
cx+ dy= 2
Consider the case when
ad- bc \neq 0
The attempt at a solution
Like in my other post, I multiplied the first equation by "c" and the second equation by "a", and then I subtracted the two equations. I...
Homework Statement
Find the general solution to the system:
ax+ by= 0
cx+ dy= 0
Consider the case when
ad- bc\neq 0
The attempt at a solution
I multiplied the first equation by "c" and the second equation by "a", and then I subtracted the two equations.
I got the following...
Whenever I am stuck I usually manage by sitting down and working on the problem and eventuall finding the solution, this one is bothering me too much and I don't have any class until friday so no hope of finding out before then unless I ask here.
Q: Find a general solution to the diff.eq...
Homework Statement
\frac{d^{2}y}{dt} +4\frac{dy}{dt}+20y=e^{-2t}(sin4t+cos4t)
Homework Equations
The Attempt at a Solution
The solution to the homogeneous equation: \frac{d^{2}y}{dt} +4\frac{dy}{dt}+20y=0 is
y= k1e^{-2t}cos4t +k2e^{-2t}sin4t
Then I guessed ae^{-2+4i} as a...
Morning everyone,
Studying for a test and having a problem on a practice question he gave us to study with. Here's the question along with the answer:
Y' = AY + [e^t
e^-t
0]
with A =
[-1 0 4
-0 -1 2
0 0 1]
the...
Homework Statement
(D^2 + 2D + 10)^2 * (D^2 - 2D -3)y = 0.
Homework Equations
D = d/dx
The Attempt at a Solution
Solving for the roots gives:
-1 + 3i, -1 - 3i <== both of multiplicity 2
and 3, -1.
So the general solution should be:
y = Ae^(3x) + Be^(-x) +...
General solution of initial value problem --dont understand problem is asking me??
Homework Statement
Find a value for y-sub-0 for which the solution of the initial value problem:
y' - y = 1+ 3sin t y(0) - y-sub-0
remains finite as t approaches infinity.
(i called it "y-sub-0" , just...
Homework Statement
This was a quiz problem
Give the value of n and C for which y=Cx^n is a solution of the equation
xdy/dx - 6y = 0Homework Equations
ans
n= 6
C= 0 (some triangle symbol) an n .. ineligible mark
The Attempt at a Solution
n = 6
C = All real numbers
my tutor explained it...
Hello,
The general solution of a differential equation for y'+P(x)y=G(x) is
y(x)=e^{-\int P(x)dx}[C+\int e^{\int P(x)dx}G(x)dx]
for y'+xy=x
y(x)=e^{-\int xdx}[C+\int e^{\int xdx}xdx] i have
y=Ce^{-\frac{x^2}{2}}+1
By the other solution
\frac{dy}{dx}+xy=x \rightarrow...
Hi,
I came across the following differential equation:
\sqrt{1+(y')^2}=\frac{d}{dx}\left(y\frac{y'}{\sqrt{1+(y')^2}}\right)
I found possible solutions: y\left(x\right)=cosh(x+C_{1}).
However, this is a second order ODE so there exist a more general solution, with 2 freedom degrees...
Hi, I am just looking for clarification whether or not I am doing these problems correctly, here's an example:
Homework Statement
Find the general solution to
x'' -2x' + 5x = 0
Homework Equations
Charasteristic polynomial.
Quadratic equation.
General solution form for complex...
Homework Statement
Find the general solution of the system of differential equations
x'=10x - 12y
y'=25x - 30y
(where primes indicate derivatives with respect to t) by using the initial conditions
x(0)=A
y(0)=B
Homework Equations
The Attempt at a Solution
x''=10x' - 12y'...
Homework Statement
x''+13y'-4x=6sint , y''-2x'-9y=0
The Attempt at a Solution
I am not really sure how to solve this completely, but I have done this so far:
(D^2-4)x + 13Dy - 6sint = 0 , (D^2-9)y - 2Dx = 0
then I hit a brick wall. Any help would be appreciated, thanks.
Homework Statement
Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the solution.
4y^{''} - 4y^{'} + y = 0
e^{x/2}, xe^{x/2}
Homework Equations...
Homework Statement
Given that \nabla2 1/r = -4\pi\delta3(r)
show that the solution to the Poisson equation \nabla2\Phi = -(\rho(r)/\epsilon)
can be written:
\Phi(r) = (1/4\pi\epsilon) \int d3r' (\rho(r') / |r - r'|)
Homework Equations
The Attempt at a Solution
I know...