Detailed Explanations for Calculus 2 Equations

Application of Rolle's
Suppose q is a nonzero function of a real-variable such that
$u^{2} q^{″} (u) + u q^{'} (u) = u^{2} q (u) + q (u)$ for all u.

Yuliana Jordan 2022-03-17

What is the frequency response of a first-order system of ODEs?
What is the frequency response of a first-order system of ODEs? Specifically, given the differential equation:
$y^{'} (t) = A y (t) + x (t)$
where y: $y : [0, \infty] \to R^{m}$ and $A \in R^{m \times m}$ , what is the solution when
$x (t) = [\begin{matrix} 0 \\ ⋮ \\ \cos (ω t) \\ ⋮ \\ 0 \end{matrix}] = \cos (ω t) b_{i} ?$
Essentially, if we have a first-order system of ODEs and we activate a single input with a sine wave of a given frequency, what is the solution? Normally, the frequency response is worked out for a linear time invariant system of higher order and the solution is just a phase shifted and amplitude scaled version of the sine wave. I'm interested in what this result looks like for a first-order system rather than a single equation.

Kayden Chandler 2022-03-17

Choice of particular solution to this second order, inhomogeneous differential equation
$y^{″} + 2 y^{'} + 5 y = x e^{x} .$
And I'm a bit confused what choice to take as my particular solution $y_{p}$ . I see that the solution to the corresponding homogeneous equation is $y_{h} (x) = e^{- x} (a \cos (x) + b \sin (x))$ .
$y_{p} (x) = A x e^{x},$
which led me to this:
$A e^{x} (x + 2) + 2 A e^{x} (x + 1) + 5 A x e^{x} = x e^{x}$ ,
which clearly cannot be solved for A.
So the next things I want to try is:
$y_{p} (x) = x e^{x} (A \cos (x) + B \sin (x))$
and then solve for both A,B. But I'm not intuitively clear why I need two unknown constants A,B instead of just one, namely A which failed.

Neveah Stewart 2022-03-17

Can 2 different ODE's have the same set of solutions?
If I have two differents linear ODE's:
$x^{''} + p (t) x^{'} + q (t) x = f (t) . p, q \in C (I, \infty) .$
$x^{''} + j (t) x^{'} + g (t) x = h (t) . j, g \in C (I, \infty) .$
Coul they have exactly the same set of solutions?
And if we have two different non-linear ODE's could they?

Paityn Nielsen 2022-03-16

Assessing stability or instability of a system of equations with complex eigenvalues
Having this system , we get clearly two complex eigenvalues. If one has to assess the stability of the system at these eigenvalues, we have for the matrix A:
$(\begin{matrix} a & b \\ c & d \end{matrix})$
that $T^{2} - 4 Δ \geq or \leq or = 0$ , where $T = a + c$ , while $Δ = a d - b c = D e t A$ . But with complex eigenvalues $\pm 2 i$ , T which must be real, becomes complex and $δ$ is always 0, when it would vary from greater than or lesser than zero for real eigenvalues. How do we solve this with complex eigenvalues?

Charity Davies 2022-03-16

Can we solve 1st order q-differential equations using the usual methods of 1st order differential equations? For example, can we use integration factor method to solve this q-differential equations?
${_qy(x)=a(x)y(qx)+b(x}_{q} y (x) = a (x) y (q x) + b (x)$

eketsaod7 2022-03-16

Calculate maximal interval by bounding $f (t, x)$
$x^{'} = f (t, x)$
$x (t_{0}) = x_{0}$

Riley Quinn 2022-03-15

When solving a first order differential equation
$\frac{d y}{d x} = f (y)$
for the fixed points (or steady states), we set the differential equation to 0 and solve for the values of y that are fixed points. Why does this guarantee that we will obtain fixed points? Why can't we obtain a point at which the slope is 0 but is not a fixed point?

ransomedv37 2022-03-15

Are there any generalisations of the identity $\sum_{k = 1}^{n} k^{3} = {(\sum_{k = 1}^{n} k)}^{2}$ ?

e3r6a2n1dz60 2022-03-15

Are there any other ways to demonstrate that
$\sin (x) = \sum_{k = 0} \frac{{(- 1)}^{k} x^{1 + 2 k}}{(1 + 2 k)!}$

Dax Gross 2022-03-15

Argue for formula about one-dimensional harmonic oscillator
$L (x, v) = \frac{m}{2} v^{2} - \frac{k}{2} x^{2}, k > 0$

Ashlynn Rhodes 2022-03-15

Applying Bendixson-Dulac Theorem in circle with radius 3
Show that the system $\frac{d x}{d t} = y + \frac{1}{5} x^{5} - x^{3},;;; \frac{d y}{d t} = - x + \frac{1}{3} x^{2} y^{3}$
has a centre at the origin, but that there are no closed orbits lying inside the circle whose equation is $x^{2} + y^{2} = 3$ .
I have shown there is a centre at the origin, but am struggling to find an appropriate function to use in the Bendixson-Dulac theorem. Any help would be much appreciated, thank you.

Pelizzolaf40 2022-03-15

Analytical solution for a separable scalar nonlinear ODE
$\dot{x} (t) = - x (t) \sqrt{1 + x {(t)}^{4}} with x (0) = x_{0} \in R$

Aiden Delacruz 2022-03-15

Alternative argument to show that function diverges everywhere
Consider the function:
$f (x) = x^{\frac{1}{2}} + \frac{1}{2} x^{- \frac{1}{2}} + \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{4} x^{- \frac{3}{2}} + \frac{1}{15} x^{\frac{5}{2}} + \dots$
which is constructed such that f is (or at least looks like) a solution to $f^{'} = f$ .
Then, $f (x) = A e^{x}$ for some constant A. But f diverges at $x = 0$ , so $A = \pm \infty$ " and f diverges everywhere.

melalcolicaoh3 2022-03-15

What is the general solution of
$\frac{d f (x)}{d x} = f (x - a)$ ?

arrostetzwyl 2022-03-14

What is the sum of the series $\sum_{k = 1}^{\infty} - \frac{1}{2^{k} - 1}$

Kassandra Cherry 2022-03-13

Where do these other initial conditions come from?
I have the problem $2 \frac{d y}{d t} = y (y - 2), y (0) = 3$ , which I get how to solve up until the intial conditions. The solution is $y (t) = \frac{2}{1 - C e^{t}}$ . The solution then plugs in three initial conditions and solves for C. They do $y (0) = 1, y (0) = - 1$ , and $y (0) = 3$ and then solve each of these. Where did the other two come from?
Also, the problem asks to determine the interval of existence. This one is straightforward, but there are others where the interval is something like $(- \infty, 2) \cup (2, \infty)$ and they plug in $t = 0$ and then take only the interval that solution lies in. Why is that?
Exact wording: Identify the equilibrium (constant) solutions for this nonlinear ODE. a) identify the equilibrium solutions for the nonlinear ODE. b) Solve the IVP for the initial condition, $y (0) = 3$ . Determine the existence interval and the limit of y(t) as t approaches the endpoints of the interval.

Aiden Delacruz 2022-03-13

Why can't I have $\dot{x} (t) < 0$ ?
I'm in the very beginning of "Ordinary Differential Equations and Dynamical Systems" by Gerald Teschl and on page 9 he starts with differential equations of the form:
$\dot{x} = f (x), x (0) = x_{0}, f \in C (R)$
To solve these he quickly goes to:
$\int_{0}^{t} \frac{\dot{x} (s) d s}{f (x (s))} = t$
Letting $F (x) = \int_{x_{0}}^{x} \frac{d y}{f (y)}$ he points out that any solution must satisfy that $F (x (t)) = t$ . So, since F(x) is is strictly monotone in some neighborhood of $x_{0}$ , any solution $ϕ$ will need to satisfy
$ϕ (t) = F^{- 1} (t), ϕ (0) = F^{- 1} (0) = x_{0}$
Assuming, $f (x_{0}) > 0$ we can find a maximal interval $(x_{1}, x_{2}) n i x_{0}$ where f is positive in this interval as well. So we can define:
$T_{+} = lim_{x ↑ x_{2}} F (x) \in (0, + \infty]$
$T_{-} = lim_{x ↓ x_{2}} F (x) \in [- \infty, 0)$
So, $ϕ \in C^{1} (\begin{array}{cc} T_{-} & T_{+} \end{array})$ . So far, so good. But, where I get confused is the following statement. Teschl states that:
In particular, $ϕ$ is defined for all $t > 0$ if and only if:
$T_{+} = \int_{x_{0}}^{x_{2}} \frac{d y}{f (y)} = + \infty$

Alexis Garner 2022-03-12

What would be the radius of convergence of $\sum_{n = 0}^{\infty} z^{3^{n}}$ ?

Shyla Singleton 2022-03-11

Which values for a will make series convergent or divergent?
$\sum_{k = 1}^{\infty} \frac{1}{(2 k + 1) \ln^{a} (2 k + 1)}$

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