Detailed Explanations for Calculus 2 Equations

Zeroes of ODE in normal form
I need to find the number of zeroes of an ODE:
$y q (x) y = 0$
when $q (x) > 0$ but monotonic decreasing. For instance:
$y \frac{y}{x^{2}} = 0$
where $x \in [1, \infty)$ .
How can I use the Strum-Comparision Theorem to estimate the number of zeroes of it's solution?
(P.S. Here the number of zeroes won't be infinitely many contrary to the general case of $q (x) > 0$ and monotonic increasing.)

Abbie Edwards 2022-03-08

Write new boundary conditions to a system of ODEs
Suppose we have the following system of equations:
${\begin{cases} {\ddot{x}}_{1} = f (x_{1}, x_{2}) \\ {\ddot{x}}_{2} = g (x_{1}, x_{2}) \\ x_{1} (0) = α \\ x_{2} (0) = β \\ {\dot{x}}_{1} (0) = γ \\ {\dot{x}}_{2} (0) = δ \end{cases}$
where all the derivatives are made in a time variable t.
Since I would like to work with a system of first order equations, I would go like this:
${\begin{cases} {\dot{x}}_{1} = x_{3} \\ {\dot{x}}_{3} = f (x_{1}, x_{2}, x_{3}, x_{4}) \\ {\dot{x}}_{2} = x_{4} \\ {\dot{x}}_{4} = g (x_{1}, x_{2}, x_{3}, x_{4}) \end{cases}$
How can I apply the starting conditions at time $t = 0$ at the new system?

Avery Velasquez 2022-03-07

Wro
ian Differential Equation.
$3 y^{″} + (\frac{6}{x}) y^{'} + 3 e^{x} y = 0$ and two $y_{1}, y_{2}$ are two partial solutions of such that $W (y_{1}, y_{2}) \neq 0$ .
(where $W (y_{1}, y_{2}) = W (x)$ is the Wro
ian of $y_{1}, y_{2}$ ). If it is known that $W (1) = 2$ , calculate W(10). I know that i have to find a function first but i do not know how. Also i can check if they are linearly independent but how can i calculate W(10). I can not find any solved examples.

Caroline Carey 2022-03-07

Wro
ian of vector
${\vec{Y}}^{'} = A Y, A_{2 \times 2}$ (constants over $R)$
${\vec{Y}}^{(1)} (x) = (\begin{matrix} e^{2 x} \\ - e^{2 x} \end{matrix}), W [{\vec{Y}}^{(1)}, {\vec{Y}}^{(2)}] (2) = e^{3}, W [{\vec{Y}}^{(1)}, {\vec{Y}}^{(2)}] (1) = e^{2}$ (W:=wro
ian).

Brittney Cuevas 2022-03-02

A bit rusty on this stuff. The whole problem is proving this is true:
$2 \leq 1 + \sum_{m = 1}^{n} \frac{1}{m!} \leq 1 + \sum_{m = 1}^{n} \frac{1}{2^{m - 1}} < 3$

ichonlarry 2022-03-01

what is the differential equation of dy/dx = x e^y

Ayden Case 2022-03-01

4th order linear ordinary differential equation?

Beverley Rahman 2022-02-28

What is the convergence rate of the tail of the series
$a_{n} = \sum_{k > n} exp (- p k) k {(\log k)}^{2}$

simnelsbl4 2022-02-28

What is the idea to sum this series of powers?
$\sum_{n = 1}^{+ \infty} (n^{2} + n + 1) x^{n - 1}$

Guy Rollins 2022-02-28

Can be series $\sum_{r = 0}^{\infty} \frac{e^{- r}}{1 + r}$ be further simplified?

Lubna Boyce 2022-02-28

I am given the series
$\sum_{n = 1}^{\infty} \frac{n^{n - 1}}{{(2 n^{2} + n + 1)}^{(n + \frac{1}{2})}}$

Scoopedepalj 2022-02-28

Why is:
$\sum_{k = 1}^{[\frac{n}{2}]} \sin^{2} ((2 k - 1) \frac{π}{n}) = \frac{n}{4}$

Cormac Gilbert 2022-02-28

Check the convergence or divergence of the series
$\sum_{n = 2}^{\infty} {(- 1)}^{n} (\frac{1}{\log n} + \frac{\cos n}{n^{2}})$

trasdulaive 2022-02-28

I have the following series:
$\sum_{n = 1}^{\infty} (2^{\frac{1}{n}} + 2^{- \frac{1}{n}} - 2)$

beuz89100g6c 2022-02-28

Prove: the complex series $\sum_{n = 0}^{\infty} (\frac{z^{n + 1}}{n + 1} - \frac{2 z^{2 n + 3}}{2 n + 3})$ is discontinuous at $z = 1$ .

Vofteldetgpt 2022-02-28

Calculate:
$\sum_{n = 0}^{\infty} n {(\frac{4}{5})}^{n + 1}$

Theresa Chung 2022-02-28

Let $f : R \to R$ be continuous and bounded. Prove that for each $x > 0$ we have
$e^{- n x} \cdot \sum_{k = 0}^{\infty} \frac{{(n x)}^{k}}{k!} f (\frac{k}{n}))$

Bradlee Hooper 2022-02-27

How do you prove that
$\frac{10^{n} - 1}{10^{n} - 2} = 1 + \sum_{i = 0}^{\infty} 2^{i} \times 10^{- n (i + 1)}$

ymhonnolfq 2022-02-27

How to prove:
$π^{2} = 18 \sum_{n = 0}^{\infty} \frac{n! n!}{(2 n + 2)!}$

Loretta Cannon 2022-02-27

Is the series
$\sum_{n = 1}^{\infty} \frac{1}{n} (\sum_{k = 1}^{n} \frac{1}{k} {(\frac{1}{2})}^{n - k})$

When you are dealing with any Calculus 2 homework, it is vital to have a look at the various questions and answers that will help you see whether you are correct in your approach to finding solutions. Even if you are dealing with analytical aspects of Calculus 2, it will be helpful as you are looking at provided equations and learn how the answers relate to original questions and problems specified.

Do not be afraid to take a look at the basic integration and related application if Calculus 2 does not sound clear or start with the Calculus 1 first.

Calculus 2

Calculus 2 Equations: Expert Solutions and More