Sydney Stanley
2022-05-02
Answered

You can still ask an expert for help

Yaretzi Odom

Answered 2022-05-03
Author has **16** answers

Step 1

Note that you have the following, which is valid for$n\ge 0$ :

$a}_{n+2}=\frac{(n+m)(n-m)}{(n+2)(n+1)}{a}_{n$

Step 2

Let$m\in \mathbb{Z}$ be any. If m is even, take $y\left(0\right)=1\text{}\text{and}\text{}{y}^{\prime}\left(0\right)=0$ , (so ${a}_{0}=1\text{}\text{and}\text{}{a}_{1}=0$ ). Therefore all the coefficients of the form $a}_{2k+1$ are zero (because of the formula above). Also, as m is an integer, ${a}_{m+2}=0$ , and as it is even for all k, ${a}_{m+2k}=0$ . So, $\text{deg}\text{}y=m$ .

Otherwise, if m is odd, let$y\left(0\right)=0\text{}\text{and}\text{}{y}^{\prime}\left(0\right)=1$ . Arguing in a similar way, you conclude that deg $y=m$ .

Note that you have the following, which is valid for

Step 2

Let

Otherwise, if m is odd, let

asked 2021-01-02

Find

asked 2022-04-01

I am interested in the following differential equation: $y{}^{\u2033}-qy$ where $q:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}^{\cdot}$ is a continuous, positive function.

asked 2021-09-13

Please, solve the differential equation. Write the method you used and solve for the dependent variable it it is possible.

asked 2022-04-01

How to solve

${\left({y}^{\prime}\right)}^{4}x-2y{\left({y}^{\prime}\right)}^{3}+12{x}^{3}=0$

asked 2022-03-31

Second-order ODE involving two functions

I am wondering how to find a general analytical solution to the following ODE:

$\frac{dy}{dt}\frac{{d}^{2}x}{{dt}^{2}}=\frac{dx}{dt}\frac{{d}^{2}y}{{dt}^{2}}$

The solution method might be relatively simple; but right now I don't know how to approach this problem.

I am wondering how to find a general analytical solution to the following ODE:

The solution method might be relatively simple; but right now I don't know how to approach this problem.

asked 2022-04-19

Solution to ${x}^{\prime}=x\mathrm{sin}\left(\frac{\pi}{x}\right)$ is unique

I want to prove that the only solution to the ODE

${x}^{\prime}=\{\begin{array}{l}x\mathrm{sin}(\frac{\pi}{x})\text{if}x\ne 0\\ 0\text{else}\end{array}$

I want to prove that the only solution to the ODE

asked 2022-04-23

Show that the change of variables $x=2u+v,y=3u+v$ transforms the linear system

$\{\begin{array}{rl}{x}^{\prime}(t)& =\phantom{3}x-2y\\ {y}^{\prime}(t)& =3x-4y\end{array}$

into

$\{\begin{array}{rl}{u}^{\prime}(t)& =-2u\\ {v}^{\prime}(t)& =-v\end{array}$

Verify that the u-axis maps to the line$y=\frac{3}{2}x$ and the v-axis maps to the line $y=x$ .

I thought for this I would just have to plug the substitutions into x'(t) and y'(t) but when doing this I receive

$\{\begin{array}{rlr}{u}^{\prime}(t)& =2u+v-2(3u+v)\phantom{\rule{-15pt}{0ex}}& =-4u-v\\ {v}^{\prime}(t)& =3(2u+v)-4(3u+v)\phantom{\rule{-15pt}{0ex}}& =-6u-v\end{array}$

into

Verify that the u-axis maps to the line

I thought for this I would just have to plug the substitutions into x'(t) and y'(t) but when doing this I receive