Detailed Explanations for Calculus 2 Equations

How can we show that $x (t) - y (t) = \int_{s - ε}^{t} f (x (r), a) - f (y (r), a (r)) d r$ implies $x (s) - y (s) = (f (x (s), a) - f (x (s), α (s))) ε + o (ε)$

Coradossi7xod 2022-03-25

How can be the state-space representation rewritten into differential equation?
${\dot{x}}_{1} = - x_{1} + x_{2}$
${\dot{x}}_{2} = - x_{1} + u$
$A (\begin{matrix} - 1 & 1 \\ - 1 & 0 \end{matrix}) x B (\begin{matrix} 0 \\ 1 \end{matrix}) u$

brunocazelles4dvb 2022-03-25

How a solving a nonlinear PDE with Self-similar Solution?
I want to solve the problem :
$\frac{\partial u}{\partial t} = {| \nabla u |}^{- 2 p} (\frac{\partial^{2} u}{\partial ξ^{2}} + (1 - p) \frac{\partial^{2} u}{\partial η^{2}})$
We search for a self-similarity solution, the general form of which is as follows
$u (x, y, t) = f (ξ), with ξ = d \frac{{(x^{2} + y^{2})}^{n}}{a (t)}$
from which we obtain
$α ξ = (1 - p) {(2 n)}^{- 2 p + 2} ((\frac{1}{2 n (1 - p)} + \frac{2 n - 1}{2 n}) {(d \frac{d f}{d ξ})}^{- 2 p} + ξ {(d \frac{d f}{d ξ})}^{- 2 p - 1} d \frac{d^{2} f}{d ξ^{2}})$
Now, I am very confused on how to solve the above equation and find the exact solution of $f (ξ)$ , thereby finding the exact solution of $u (x, y, t)$ . So my question is, does anyone know how to solve this ordinary differential equation?

Kason Palmer 2022-03-25

Given a function f with $f (1, 940) = 1, 000, 000$ , and its derivative is equal to $k f (200 - f)$ and k is a constant. How can I get f?

ICESSRAIPCELOtblt 2022-03-25

From the equation
$(F_{x} + p F_{y}) d x + F_{p} d p = 0$
I do not understand how we can get
$\dot{x} = F_{p}, \dot{p} = - (F_{x} + p F_{y})$

Aryan Salinas 2022-03-24

How do I compute the maximum value of the solution of this differential equation?
${\begin{aligned} \frac{d y}{d t} & = f (y) = 1 + k ϵ - k y - e^{- y} \\ y (0) & = ϵ \end{aligned}$

Coradossi7xod 2022-03-24

How could we prove this ?
$\sum_{i = 1}^{n} (i^{2} + 3 i + 1) \times i! = (n + 3) \times (n + 1)! - 3$

Nancy Richmond 2022-03-24

Help needed solving logistic differential equation with initial conditions
$\frac{d P}{d t} = P (10 - 2 P)$
with initial conditions of $P (0) = 1 .$

alparcero97oy 2022-03-24

Help to find the solution of
$y^{'} = (\begin{matrix} - 2 & 1 \\ 1 & - 2 \end{matrix}) \cdot y + (\begin{matrix} 2 \sin (t) \\ 2 (\cos (t) - \sin (t)) \end{matrix})$

k4s3thg368 2022-03-24

Given: $y^{2} = c e^{x} - x - 1$
Find the orthogonal family of curves and write 2 curves from both families that pass in (2,0).

ikramkeyslo4s 2022-03-24

Limit of a separable equation:
Consider the equation
$\frac{d y}{d t} = k (a - y) (b - y)$
where a,b and k are constants. Assuming $y (0) = 0$
a) Solve for y(t) when $a = b$
b) Solve for the case $0 < a < b$
c) By considering the limit $b \to a$ in (b) show that the two results are consistent

Jazmyn Holden 2022-03-23

Solve the heat equation using a transform method
$k \frac{\partial^{2} U}{\partial x^{2}} = \frac{\partial U}{\partial t}$
subject to
$U (0, t) = 1, t > 0$
$U (x, 0) = e^{- x}, x > 0$

Addison Fuller 2022-03-23

Solve $3 y^{'} + 6 x y = 6 e^{- x}$ where $y (0) = 1$ .
I am trying to solve $3 y^{'} + 6 x y = 6 e^{- x}$ where $y (0) = 1$ . This is a first-order linear equation. Dividing by 3 and multiplying by $e^{x^{2}}$ , we have
$e^{x^{2}} y^{'} + 2 x e^{x^{2}} y = 2 e^{x^{2} - x} .$
Thus, we have $\frac{d}{d x} (e^{x^{2}} y) = 2 e^{x^{2} - x}$ .
Now, we have $e^{x^{2}} y = 2 \int e^{x^{2} - x} d x$ .
How do I calculate this integral? The solution is supposedly $y (x) = (2 x + 1) e^{- x^{2}}$ .

resaplaha7dl 2022-03-23

Limit for $t r i g h t a r r w o \infty$ of a solution to an ODE
Consider the Cauchy problem:
${\begin{cases} y^{'} (t) = \frac{t^{2}}{2 + \sin (t^{2})} c o s (y (t)^{2}) \\ y (0) = 0 \end{cases}$

basura8w081 2022-03-23

Let's say for example these two:
$\frac{3 n}{2 n + 3} {(- 1)}^{n + 1} - \frac{{(- 1)}^{n}}{2 n} {({(- 1)}^{n} + \frac{1}{n})}^{2}$

ezpimpin6988ok1n 2022-03-23

Let $f : R \to R$ be a differentiable function, and suppose $f = f^{'}$ and $f (0) = 1$ . Then prove $f (x) \neq 0$ for all $x \in R$

meli199939f 2022-03-23

Let N be a positive integer. Find all real numbers a such that the differential equation $\frac{d^{2} y}{d x^{2}} - 4 a \frac{d y}{d x} + 3 y = 0$ has a nontrivial solution satisfying the conditions $y (0) = 0$ and $y (2 N π) = 0$

Nadia Clayton 2022-03-22

Solve this nonhomegenous ode
$y^{″} + 4 y = \cos (2 x)$

juctommaccedo662f 2022-03-22

Solve this equation in Sturm-Liouville.
$(1 - x^{2}) y^{″} - x y^{'} + λ y = 0$
My solution:
$μ (1 - x^{2}) y^{″} - μ x y^{'} + μ λ y = {(p y^{'})}^{'} + (λ r - q) y$
$μ (1 - x^{2}) y^{″} - μ x y^{'} + μ λ y = p^{'} y^{'} + p y^{″} + (λ r - q) y$
$p^{'} = μ x, p = μ (1 - x^{2}) \Rightarrow \frac{p^{'}}{p} = \frac{x}{1 - x^{2}} \Rightarrow \ln (p) = - \frac{1}{2} \ln | 1 - x^{2} | \Rightarrow p = \frac{1}{\sqrt{(| 1 - x^{2} |)}}$
Then $μ = \frac{1}{{(1 - x^{2})}^{3}}$
$\frac{1}{{(1 - x^{2})}^{2}} y^{″} - \frac{1}{{(1 - x^{2})}^{3}} x y^{'} + \frac{1}{{(1 - x^{2})}^{3}} λ y = p^{'} y^{'} + p y^{″} + (λ r - q) y$

juctommaccedo662f 2022-03-22

Solve the series:
$lim_{n \to \infty} \frac{k + 1}{3^{k}}$

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