# 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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### When solving a third-order ODE using reduction of order, what happens when the v’ term drops out

Given question Given $y=x$ is a solution of $6x^3y’’’ - 24x^2y’’ + 48xy’-48y = 0$. Find the general solution if $x>0$. My work Since we are given that $y = x$ is a solution, my first approach would ...
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### Frobenius method for solving differential equation

Need help solving this differential equation$$xy''+(x+l)y'+ly=0$$ where $l\in \mathbb{R}$\ I tried using the Frobenius method. $x_0=0$ is a regular singular point and Assume that we have the solution ...
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### How can I solve a Bessel equation with Reduction of order?

If $y_{1}(x) = \frac{\sin(x)}{\sqrt(x)}$ is one solution of the differential equation $$x^2y'' +xy' + (x^2-\frac{1}{4})y = 0$$ find the second solution $y_{2}(x)$. My effort using Wronskian The ...
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### Reduction of Order to solve, but can't get right answer

Given the equation y'' + 2y' + y = 0 and the solution y1 = ${xe}^{-x}$. Solve for a second solution $y_2$. I've solved it twice and get $y_2 = {xe}^{-x}(-{x}^{-1}+c)$ , but that's not being counted ...
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### Solve the ODE $y y''=3(y')^2$ using Reduction of Order

For the reduction of order method, we are supposed to guess a solution, $y_1$, and assume that the second solution (this will be second order as the title shows) is of the form $y_2=uy_1$. But I'm ...
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### Solving 3rd Order ODE by Reduction of Order

I am trying to solve the following ODE: $x^3y''' -3x^2y'' +(6-x^2)xy' -(6-x^2)y=0$ if $y_1=x$. The professor said that the solution could be found by reduction of order. If it's second order the ...
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### Why Can Reduction of Order Automatically Find the "Defective" Fundamental Solution

To compute generalized eigenvectors of defective linear ODE systems, it is up to the mathematician's trial and error to find and plug in the specific eigenvector(s) which are in the column space of ...
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### Reduction of order on a 2nd order LI DE

The equation and $y_1(x)$ are $$y_1(x)=x^{\frac{-1}{2}}\cos(x)$$ $$x^2y''+xy'+(x^2-\frac{1}{4})y=0$$ Using the formula that solves for $y_2(x)$ (the one with an integration factor over $y_1(x)$ ...
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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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### 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 ...
1 vote
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### 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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### 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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### 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 ...
1 vote
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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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### 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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### 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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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 ...