What is Second order ode: Definition and 91 Discussions
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
I know how to solve similar ODEs like
##
\frac{\partial^2 x}{ \partial t^2} + b \frac{\partial x}{ \partial t} + C x =0
##
Where one can let ## x = e^{rt}##, and the equation becomes
##
r^2 + b r + C =0
##
Which can be solved as a quadratic equation.
But now the problem is that there is...
I have found that w(x) should be e^-x to make L self-adjoint.
and insert back get xL''+(x+1)L' +lambda L = 0
now it needs to assume a monic polynomial function, so I assume Ln = x^n+ sum from k=0 to n-1 (a_k*x^k)
get the 1st and 2nd order differential and insert back
I get lambda_n =...
I shall not begin with expressing my annoyance at the perfect equality between the number of people studying ODE and the numbers of ways of solving the Second Order Non-homogeneous Linear Ordinary Differential Equation (I'm a little doubtful about the correct syntactical position of 'linear')...
Hi all,
I have another second order ODE that I need help with simplifying/solving:
##p''(x) - D\frac{e^{\gamma x}}{A-Ae^{\gamma x}}p'(x) - Fp(x) = 0##
where ##\gamma,A,F## can all be assumed to be nonzero real numbers and ##D## is a purely nonzero imaginary number.
Any help would be appreciated!
I believe it is the case that any linear second order ode with at most 3 regular singular points can be transformed into a hypergeometric function. I am trying to solve the following equation for a(x):
where E, m, v, k_{y} are all constants and I believe turning it into hypergeometric form will...
i am new to MATLAB and and as shown below I have a second order differential equation M*u''+K*u=F(t) where M is the mass matrix and K is the stifness matrix and u is the displacement.
and i have to write a code for MATLAB using ODE45 to get a solution for u. there was not so much information on...
Question:
So I got around on doing this example, and I'm pretty sure I messed up somewhere, would appreciate if someone could point out what I did wrongly.
1) For any second ODE, I should let:
##y_{1}= y ## and ##y_{2}= y' ##
Hence,
##y_{1}'= y' = y_{2} ## and ##y_{2}'= y'' = xy(x)+x^2-y(x) =...
With the new variable, I got:
$$p^2 (p'_y)^{2}=k^2(1+p^2)$$ where ##p'_y## is ##\frac{dp}{dy}##.
I modified the equation so the variable p and dp can be separated from dy. Here what I got:
$$\frac{p}{\sqrt{p^2+1}} dp=k dy$$
I substitute ##p^2+1=u## so I got
$$\sqrt{u}=ky+c_1$$
Back substitution...
I'm trying to solve the following nonlinear second order ODE where ##a## and ##b## are constants: $$\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}-\frac{y}{ay+b}=0$$ It looks somewhat like the modified Bessel equation, except the third term on the left makes it nonlinear. I've been trying to...
I have currently been reading a book called 'Mathematical Methods In Physical Sciences'. Whilest I was looking at the differential section I came across a differential which I have never thought about before, which is of the form...
Homework Statement
The question I am working on is the one in the file attached.
Homework Equations
y = u1y1 + u2y2 :
u1'y1 + u2'y2 = 0
u1'y1' + u2'y2' = g(t)
The Attempt at a Solution
I think I have got part (i) completed, with y1 = e3it and y2 = e-3it. This gives a general solution to the...
Homework Statement
Hi there, I have an assignment which involves using reduction of order to solve for a second solution to an ode (the one attached). However this is a method I am new to, and though I have tried several times, I'm somehow getting something wrong because the LHS and RHS are not...
Homework Statement
d2u/d2x + 1/2Lu = 0 where L is function of x
Homework Equations
I am try to find solutions y1 and y2 of this equation.
The Attempt at a Solution
y = [cos √(L/2) x] + [sin √(L/2) x]
y' = - [√(L/2) sin √(L/2) x] + [ √(L/2) cos √(L/2) x]
y'' = -[(L/2) cos √(L/2) x] -...
I have this second order differential equation but I'm stumped as to how to solve this since the zeroth order term has a Sine function in it and the variable is embedded.
##\ddot y(t) + 3H (1+Q) \dot y(t) -m^2 f \sin(\frac{y(t)}{f}) = 0##
##H~##, ##~Q~##, ##~m~##, and ##~f~## are just...
Homework Statement
I have this set of equation:
My''+Cy'+Ky=0 but C=0
M is a matrix consist of {(-m) (0)/( -1/12mb^2) (-1/12mb^3)}
and K is a matrix of {(-K1-K2) (-K2b)/ ((K1b-K2b)/(2)) (-K2b^2/2)}
and y is a coordinate system which is (x1,θ)
Now i have to convert these...
Homework Statement
The harmonic oscillator's equation of motion is:
x'' + 2βx' + ω02x = f
with the forcing of the form f(t) = f0sin(ωt)The Attempt at a Solution
So I got:
X1 = x
X1' = x' = X2
X2 = x'
X2' = x''
∴ X2' = -2βX2 - ω02X1 + sin(ωt)
The function f(t) is making me doubt this answer...
Homework Statement
I need to solve:
x^2y''-4xy'+6y=x^3, x>0, y(1)=3, y'(1)=9
Homework EquationsThe Attempt at a Solution
I know that the answer is: y=x^2+2x^3+x^3lnx
Where did I go wrong. I was wondering if it's even logical to solve it as an Euler Cauchy and then use variation of parameters...
The second order ODE is,
\begin{equation*}
\frac{d^2 x}{dt^2} = -\omega^2_g \frac{dx}{dt}
\end{equation*}
I tried solving this by substitution of the second order derivative into a variable and transforming the equation into a second order polynomial, and I get the solution involving an...
I'm supposed determine whether following statements are true or false. However, I can't get past the notation.
Question: the second order differential equation $\ddot{x}+\dot{x}+x = 9t$ is:
(a) equivalent to $\begin{cases} \dot{x} = y, & \\ \dot{y}=-y-x+9t, &\end{cases}$ (b) solved by...
I would like to solve the steady-state one dimensional heat equation for a two piece material system. The thermal conductivity in each segment is a linear function of temperature, where ##\kappa_1=a_1T+b_1## for material 1 and ##\kappa_2=a_2T+b_2## for material 2. ##a_1, a_2, b_1, and \;b_2##...
Homework Statement
(Reduction of order) The function y1 = x-1/2cosx is one solution to the differential equation x2y" + xy' + (x2 - 1/4) = 0. Use the method of reduction of order to find another linearly independent solution.
The Attempt at a Solution
I divided x2 to both sides to get the...
Homework Statement
I've been stuck on this problem for three days now, and I have no clue how to solve it.
Construct a linear differential equation of order 2, for which { y_1(x) = sin(x), y_2(x) = xsin(x)} is a set of fundamental solutions on I = (0,\pi) .
Homework Equations
Wronskian for...
Homework Statement
A weight of 8 pounds extends a spring 2 feet. It's assumed that the damping force that acts on the system is equal (number-wise) to alpha times the speed of the weight.
Determine the value of alpha > zero so x(t) is critically damped.
Determine x(t) if the weight is liberated...
Hello everyone; i'd like some help in this problem : i want to solve num this differential equation
{ y"(t)+t*cos(y)=y } by the Taylor method second order expansion. i first have to make this a first order differential equation by taking this vector Z=[y' y] then we have Z'=[y" y'] which equal...
I'm a little stuck getting started on this question. y''+\tan(x)y=e^x with y(0)=1,y'(0)=0. I know the existence and uniqueness theorem
for an nth order initial value problem
How do I apply the theorem?
Homework Statement
Find the general solution.
Homework Equations
y"+y=x2sin2x
The Attempt at a Solution
Characteristic equation would be:
m2 + 1 = 0
So,m2 = -1
Therefore, m = i or m = -i.
Complementary function would be : Asinx+Bcosx where,A and B are constants respectively.
If I write...
if I consider a circuit consisting of a capacitor, an inductor and a resistor and using kirchhoffs voltage rule for the circuit i come up with the following
L(Q''(t)) + R(Q'(t)) + (Q(t))/C = 0 I solve for the roots using a characteristic equation of the form
LM2 +MR +(1/C) = 0
solving this for...
Homework Statement
I'm taking an online introductory chem course, and while explaing the failure of classical mechanics to describe electron behavior, the teacher brought up the following ode which is based on Newton's second law and coulombs law:
-e^2/4(pi)(epsilon-nuaght)r^2=m(d^2r/dt^2)...
Homework Statement
u'' + w20*u = cos(wt)
w refers to omega.
Homework EquationsThe Attempt at a Solution
I'm not sure where to begin on this. For starters, it's a multiple choice problem, and all the answers are given in terms of y, so I'm not sure if u is supposed to replace y' or something...
Homework Statement
Solve:
\frac{d^{2}y}{dx^{2}} + \omega^{2}y = 0
Show that the general solution can be written in the form:
y(x) = A\sin(\omega x + \alpha)
Where A and alpha are arbitrary constants
Homework EquationsThe Attempt at a Solution
I know that I will need to change variables for...
Homework Statement
y'' + 4y = t2 + 6et; y(0) = 0; y'(0) = 5
Homework Equations
The Attempt at a Solution
So, getting the general solution, we have r2 + 4 = 0, so r = +/- 2i
So the general solution is yc = sin(2t) + cos(2t)
I then used the method of undetermined coefficients to figure that...
Homework Statement
y''+6y=f(t), y(0)=0, y'(0)=-2
f(t)= t for 0≤t<1 and 0 for t≥1
Homework Equations
The Attempt at a Solution
L{y''}+6L{y}=L{t}-L{tμ(t-1)} where μ(t-1) is Unit Step
Y(s)=L{y}
sY(s)-y(0)=L{y'} and y(0)=0
s2Y(s)-sy(0)-y'(0)+6Y(s) where y(0)=0 and...
Homework Statement
A mass of 5kg stretches a spring 10cm. The mass is acted upon by an external force of 10sin(t/2) Newtons and moves in a medium that imparts a viscous force of 2N when the speed of the mass is 4cm/sec. If the mass is set in motion from its equilibrium position with an initial...
I had made a post in the past about the same problem and unfortunately I wasn't clear enough
so I am trying again.
I am studying an article and there I found without any proof that the solution of:
Let ##g \in \mathbb{C}## and let ##u:(0,\infty)\to \mathbb{C}##
$$ -u''+\lambda^2u=f\,\, on...
-u''(z)+α2u(z)=f(z), u(0)=g(z), u(z)=0 as z→∞
-u''(z)+α2u'(z)=f(z), u(0)=g(z), u(z)=0 as z→∞
I am interested to solve these two boundary problems using Green's functions. It is noticed that z is complex variable. Can someone help me to do this?
Hi all,
I have a nonlinear ODE in the following form:
a x'' + b |x'|x' + c x' + d x = y
where x and y are functions of time and a,b,c and d are constants. As far as I can tell the only way to solve this is numerically, something I've managed to do successfully using a Rung-Kutta scheme...
Homework Statement
basically solve \frac{d^{2}y}{dx^{2}} + 4\frac{dy}{dx} + 4y = cos2x
Boundary conditions are y=0, dy/dx =1 at x=0
Homework Equations
The Attempt at a Solution
I am having trouble getting the coefficients to the solution. I got the complementary function as...
Homework Statement
I must solve ##y''+2y'+2y=e^{-t}\sin t##.
I know variation of parameters might not be the fastest/better way to solve this problem but I wanted to practice it as I never, ever, could solve a DE with it. (Still can't with this one). Though the method is supposed to work...
Homework Statement
Hello guys! I've never dealt with an ODE having 2 singularities at once, I tried to solve it but ran out of ideas. I must solve ##(x-2)y''+3y'+4\frac{y}{x^2}=0##.
Homework Equations
Not sure.
The Attempt at a Solution
I rewrote the ODE into the form...
Homework Statement
Find the general solution of the ODE $$ y'' + 16y = 64x \cos x.$$ If ## y(0)=1, y'(0) = 0##, what is the particular solution?
The Attempt at a Solution
I am confident I can tackle this question, I really just want to check that my particular integral form is correct. I...
Not sure if this topic belongs here, but here goes.
Homework Statement
From the AP physics C 1995 test there is a problem that gives the potential energy curve U(x). With F=-\frac{dU}{dx} in one variable,
F(x)=-\frac{a}{b}+\frac{ba}{x^{2}}
Where a and b are constants. Now I need to get...
I was given the following equation to solve:
x^2*y'' + x*y' + k^2*x^2*y = 0
B.C. y'(0)=0, y(1)=0
where k is just some constant.
I am having a hard time removing the singularity created by the boundary condition at y' and not aware of a method how. Any advice would be greatly appreciated.
Homework Statement
I'm pretty sure this is a typo?
http://gyazo.com/802746486cc68852e5384d5a12aed596
Homework Equations
See the image ^.
The Attempt at a Solution
I believe the theorem they're talking about, is that you can write the general solution of a second order ODE :
L[y] = y'' +...
Homework Statement
Need to solve
xy''+y'+xy=0 using Runge Kutta on x[1,3]
Couldn't find algorythm to solve second order ODE using this method
I know how to do 1st order
Homework Equations
The Attempt at a Solution
I know I have to make this equation into 2 first order ODE...
Help solving a second order ODE with repeated roots, urgent!
I have a differential equaition
d2y/dx2 - 6dy/dx + 9y = 0
I have found the general solution to be
y = (Ax + B)e3x
Now I need to find the solutions to A and B so that...
when y = 4, x = 0
when y = 49.e15, x = 5
I...
Hey all,
there is something that has always bugged me in linear second order ODEs. We say that the general solution is:
y=C_1e^{r_1x}+C_2e^{r_2x}
where r_1 and r_2 are the solutions of the characteristic polynomial.
The cases where r1, r2 are real are pretty straightforward. If they are...
I'm not sure exactly how to solve this ODE. (dx^2)/(dt^2) + (w^2)x = Fsinwt, where x(0) = 0 and X'(0) = 0.
What I've got so far is:
x'' + w^2x = Fsinwt --> x(homogenous) = Acoswt + Bsinwt
I know I have to find a particular solution but I keep getting zero as a result which I know won't...