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Questions tagged [reduction-of-order-ode]

To use when our aim is to solve an ODE (of any order) only with reduction of order method. If you just want to solve an ODE use the tag differential-equations.

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I need some help finding a second solution to $t^2 y''-2ty=3t^2-1$ given the fact that $y_1=t^{-1}$

Using the fact that $y_1$=$t^{-1}$ then $y_2$= z $\cdot$ $t^{-1}$ $y_2'= z'\cdot t^{-1} - z \cdot t^{-2}$ $y_2''= z''\cdot t^{-1} - z'\cdot t^{-2} - z'\cdot t^{-2} + 2 \cdot z \cdot t^{-3}$ ...
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Finding the roots of a characteristic polynomial

Main aim is to find the lowest order equation with the solution: $$y(x)= 2 \cosh(x) + 3 e^{-2x} \sin(x)$$ Now, I am trying to find the roots to form the characteristic polynomial from which I get the ...
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Halting problem proofs and reduction

Okay so Ive got a problem with the "direction" of reduction and untimately the whole proof, reduction idea as a whole... My Question is: Prove that the problem of deciding whether a Turing Machine ...
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Reduced order model from time series to generate a new time series

I am working on simulation and created a model made of many equations. The simulation takes very long time to simulate and make results. So my aim is I want to be able to speed up the process using ...
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1answer
70 views

Doubt on differential equation involving the complementary error function

How to find the general solution of $y''+2xy'-2ny=0$? I was solving a problem in this thread. The posted solution involves the error function which I am not aware of. I found this link. How to find ...
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20 views

Proving NP-hardness of GAP problem using PCP theorem

I want to prove that for $0<\epsilon$ , the problem GAP 2-SAT[$\frac{11}{16}$ + $\epsilon$, $\frac{7}{8}$] is NP-Hard. According to the PCP theorem, the problem GAP 3-SAT[$\frac{7}{8}$ + $\epsilon$...
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253 views

Reduction of Order Leads to Non-Elementary Integral

If $u_1=x+1$ is a solution of $$xu''-(x+1)u'+u=0$$ find another linearly independent solution using reduction of order. I let $u_2=(x+1)v(x)$ be the second solution. Hence $$v''(x^2+x)-v'(x^2+1)=0$$ ...
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Reduction of Order via Substitution

Suppose $u_1=\sin{x^2}$ is a solution of $$xu''-u'+4x^3u=0\Rightarrow u''-x^{-1}u'+4x^2u=0 \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ I am trying to find a second, linearly independent, solution (say $u_2$)...
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251 views

Find a solution to the differential equation: Which is NOT a constant multiple of the solution given.

Find a solution to the equation: $$ ty'' - (t+1)y' + y =0 $$ Which is not a constant multiple of the solution: $y(t) = e^t$. Hi, I have worked this problem and have solved it, using the particular ...
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Using reduction of order to find general solution of DE

I have been given the following DE... $$ ty' ' - ( 1 + 3 t ) y ' + 3 y = 0 \ , \ \ t \neq 0 $$ The problem states that the DE has a solution of form $ e^{ct} $, where $c$ is some constant. ...
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Relevant resources of reduction algoritm Generalized Mersenne primes, Proth primes, Fermat primes and Barrett reduction algorithm with sample examples

I want to get some resources about reduction algorithm of Generalized Mersenne primes, Proth primes, Fermat primes (lazy reduction) with sample examples and Barrett algorithm with sample examples. I ...
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2answers
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Solving a DAE (differential algebraic equation)

I'm interested in differential algebraic equations. [https://it.mathworks.com/help/matlab/math/solve-differential-algebraic-equations-daes.html] I have this one: $$y_1'(t)=-y_2(t)$$ $$0=y_1(...
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637 views

Solve $x^2y''-xy'+y=0$ given that $y_1=x$ using reduction of order

Hello so I need to solve $x^2y''-xy'+y=0$ given that $y_1=x$ using the idea that $y_2=v(x)x$. So I went through all the steps $$ x^3v''+2x^2v'-xv-x^2v'+xv = x $$ $$ x^3v''+x^2v' = x $$ $$ xv'' + v' =...
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1answer
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Reduction of order where $y_1=(\beta \tan^2x+1)$ with $\beta \in R$

I need to apply the reduction of order into differential equation $$\cos^2xy''-6y=0$$where the first solution is of the form $$y_1=(\beta \tan^2x+1)$$ with $\beta \in R$. I started solving equation ...
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68 views

Find a reduction formula for the following:

$$I_n = \int\frac{x^{2n}}{(1+x^2)^{1/2}}$$ Having difficulty with, $$x^{2n}$$ oppose to the more common, $$x^{n}$$ which can be split into: $$x^{n-1}*x$$ Used integration by parts to achieve the ...
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1answer
244 views

Reduction of order for a third order differential equation

Given $y'''(x) + P(x)y''(x) + Q(x)y'(x) + R(x)y(x) = 0$ with $P, Q$ and $R$ continuous functions and a solution $y_1(x)$ of the equation, how can we determine two other solutions. I used reduction of ...
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Reduction of Order Non-homogenous System [closed]

I have to write the following system as a first order system of differential equations x'''-2x''+4x'-y= sin(t) y''-2y+x''= cos(t) I have no idea where to start.
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Find initial expression that a reduction process with 2 rules does not terminate

Imagine a language of finite sequences of 0 and 1. The rules for simplifying strings in this language are given by: l??x => x1101 0??x => x00 In these rules, the variable x denotes an ...
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rectification of direction field and coordinate system

Consider $$ \dot x=0,\qquad \dot y =1 \qquad (\star) $$ Rectify the induced direction field is to find a diffeomorphism $(\tau,u,v) =\phi(t,x,y)$ such that $\frac{d u}{d \tau}=0=\frac{d v}{d\tau}$. ...
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Verifying that $e^t$ is a solution to $(2t − t^2)y''+ (t^2-2)y'+2(1-t)y=0$, then deriving the general solution.

Verify that $e^t$ is a solution of the differential equation $$(2t − t^2)y''+ (t^2-2)y'+2(1-t)y=0.$$ Use this solution to find the general solution.
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Recasting $\phi_{tt}(x,t)=\phi_{xx}(x,t)$ in “first-order form”

Consider the basic one-dimensional wave equation $$\phi_{tt}(x,t)=\phi_{xx}(x,t)$$ with boundary conditions taken to be the Sommerfeld radiation conditions: \begin{align} \phi_t(0,t)&=\phi_x(0,...
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Is it possible to reduce a LQG into a PID?

Assume that we have a LQG controller and our state space model is known. Now we are implementing an integral action into the LQG controller, so it would be a LQGI controller. The LQGI controller are ...
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1answer
72 views

Integration and Absolute Values

I am doing a Reduction of Order problem that requires an Integrating Factor. I understand how to do it [almost] perfectly fine. I do not understand a couple steps in the integration processes. $$xy'...
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1answer
37 views

Integrating an exponential function with a product?

The original problem I was tasked with solving is $$(x-1)y'' -xy' + y = 0$$ and given the solution $y_1(x) = e^x$. I started working through the problem using this method (http://www.sosmath.com/...
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3answers
67 views

Can I absorb the sign into the constant of a solution to the homogeneous equation in a second order ode?

I want to solve $$y''+\frac{x}{1-x}y'-\frac{1}{1-x}y=(x-1)e^{x}$$ using variation of parameters. Now, I could guess a solution to the inhomogeneous, i.e. $y_1 = x$. Then I used the formula of ...
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1answer
347 views

Reduction of order for third order ode

This question is similar to this question, however I do not understand the method I should be using. I need to solve $$x^3y'''+7x^2y''+xy'-16y =0 \,\,\,\,\,\,\, \text{knowing that} \,\,\,\,\,\ y_1 = x^...