Detailed Explanations for Calculus 2 Equations

Pendulum with Dirac Comb excitation
$l \ddot{θ} + b \dot{θ} + g θ = G, \sum_{- \infty}^{\infty} δ (t - n T)$
where l, b, g, G are constants and $T = \frac{2 π}{ω}$ .
Show that the resulting motion is given by
$θ (t) = \frac{G}{T l ω^{2}} + \frac{2 G \cos (ω t - \frac{π}{2})}{T b ω} +$ [terms with frequencies $\geq 2 ω$ ]
and explain why the higher frequency terms are supressed.
My first take was to rearrange to
$\ddot{θ} + \frac{b}{l} \dot{θ} + ω^{2} θ = \frac{G}{l}, \sum_{- \infty}^{\infty} δ (t - n T)$
where $ω = \sqrt{\frac{g}{l}}$
Then, taking the Laplace transform of both sides I got
$Θ (s) = \frac{G}{l, \sqrt{ω^{2} - {(\frac{b}{2 l})}^{2}}}, \frac{\sqrt{ω^{2} - {(\frac{b}{2 l})}^{2}}}{{(s + \frac{b}{2 l})}^{2} - ({(\frac{b}{2 l})}^{2} - ω^{2})}, \frac{1}{1 - e^{- s T}}$
which, as far as I'm concerned transforms to
$θ (t) = \frac{G}{l, \sqrt{ω^{2} - {(\frac{b}{2 l})}^{2}}} \sum_{n = 0}^{\infty}, H (t - n T), e^{- \frac{b}{2 l} (t - n T)} \sin (\sqrt{ω^{2} - {(\frac{b}{2 l})}^{2}} (t - n T))$
And, assuming that this is a correct form of the solution, I can't see how that is equivalent with the function given in the question. I reckon it has something to do with using Fourier series/transform instead? If so, I'm not sure how to do that. Or, is there a way to convert my solution into the given one?
I've been struggling with this for a good few days now, so any help would be much appreciated.

Travis Caldwell 2022-04-23

Nonlinear ODE initial value problem
A student came to me with a problem I couldn't solve. It's the beginning of the semester in his Intro DiffEq class, and so the solution shouldn't be too difficult. But it completely stumped me, and now I can't let it go! Here it is:
Problem. Find all solutions to the IVP:
$\sqrt{{(\frac{d y}{d x})}^{2} - 4 x^{2}} = \sqrt{x^{4} - y^{2}},;;;; y (0) = 0 .$
I thought about maybe just doing a sort of guess-and-check method, but the existence/uniqueness theorem doesn't apply, so I'm not sure that would help even if I could find a single solution, which I cannot in any case.

Jazmine Sweeney 2022-04-23

No solution to general Volterra integral equation of the second type?
I encountered a (likely simple) Volterra-style integral equation as part of my work. For an arbitrary function p(s) which is square integrable, I have the following equation relating p(s) and its integral with respect to the real-to-real function Q:
$p (x) - C \int_{0}^{x} p (s) d Q (s) = 0$
where C is a known real constant.
I would hopefully like to show, without further assumptions, that this system has only the solution $p (s) = 0$ . I have very little training in differential equations, so I'm a bit lost as where to begin.

Elise Winters 2022-04-23

Is it possible to solve this first-order O.D.E using the separable equations method?
$\frac{d y}{d x} = x - 1 + x y - y$

Wertgenib6 2022-04-23

Inhomogeneous system of DE in matrix form
$\dot{x} = 2 x + y + 3 e^{t}$
$\dot{y} = 4 x - y$

g2esebyy7 2022-04-23

Solution of nonlinear BVP is symmetric where derivative is 0:
For the following nonlinear BVP,
$y^{″} (x) + y {(x)}^{2} = 0, y (0) = y (1) = 0$ ,
I wish to show that if
$y^{'} (x_{0}) = 0$ , then $y (x) = y (x_{0} - x)$ .
That is, the solution is symmetric about this point.
I've already shown that the nontrivial solution is unique, so it should be enough to show that $z (x) = y (x_{0} - x)$ satisfies the BVP where y is a nontrivial solution but I can't seem to do this.

Dania Robbins 2022-04-23

Solution of simultaneous ODE :
$x^{'} = (x^{2} + y^{2}) y$ and $y^{'} = - (x^{2} + y^{2}) x$

dooporpplauttssg 2022-04-23

Solution of simultaneous ODE:
$x^{'} = (x^{2} + y^{2}) y and y^{'} = - (x^{2} + y^{2}) x$

znacimavjo 2022-04-23

Solution of the differential equation $, f (x) f^{'} (x) + f^{'} (x) = g (x)$ ?
How do we solve a differential equation of the form
$f (x) f^{'} (x) + f^{'} (x) = g (x) ?$
The coefficient f(x) of f'(x) post a difficulty that integrating factors doesn't work.

Angeline Hayden 2022-04-23

Solution of first order integro-differential equation
Consider $\frac{d y}{d x} + p (x) y = \int_{0}^{\infty} y (x) d x = R \approx (s a y)$
$y (0) = \int_{0}^{\infty} f (x) y (x) d x .$
I want to solve above differential equation. Here is my try:
$y (x) = y (0) e^{- \int_{0}^{x} p (τ) d τ} + R \int_{0}^{x} e^{- \int_{s}^{x} p (τ) d τ} d s .$
where $R = \int_{0}^{\infty} y (0) e^{- \int_{0}^{x} p (τ) d τ} d x + R \int_{0}^{\infty} \int_{0}^{x} e^{- \int_{s}^{x} p (τ) d τ} d s d x$
$y (0) = \int_{0}^{\infty} f (x) y (0) e^{- \int_{0}^{x} p (τ) d τ} d x + R \int_{0}^{\infty} f (x) \int_{0}^{x} e^{- \int_{s}^{x} p (τ) d τ} d s d x .$
Now, I am not able to find R which is independent of y(0) and y(0) which is independent of R. Please let me know if it is possible to find solution of this integro-differential equation.

Beedgighref28n 2022-04-23

Show that the change of variables $x = 2 u + v, y = 3 u + v$ transforms the linear system
${\begin{aligned} x^{'} (t) & = x - 2 y \\ y^{'} (t) & = 3 x - 4 y \end{aligned}$
into
${\begin{aligned} u^{'} (t) & = - 2 u \\ v^{'} (t) & = - v \end{aligned}$
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
${\begin{aligned} u^{'} (t) & = 2 u + v - 2 (3 u + v) & = - 4 u - v \\ v^{'} (t) & = 3 (2 u + v) - 4 (3 u + v) & = - 6 u - v \end{aligned}$

Kaiya Hardin 2022-04-23

Power series solution of $y^{'} - y = x^{2}$ .
I try to get a power series solution.
$y (x) = \sum_{n = 0}^{\infty} a_{n} X^{n}, y^{'} (x) = \sum_{n = 1}^{\infty} n a_{n} X^{n - 1} = \sum_{n = 0}^{\infty} (n + 1) a_{n + 1} X^{n}$
$\sum_{n = 0}^{\infty} (n + 1) a_{n + 1} X^{n} - \sum_{n = 0}^{\infty} a_{n} X^{n} = X^{2}$
Then $a_{n + 1} = \frac{a_{n}}{n + 1} \forall n \neq 2$
$n = 2 \Rightarrow 3 a_{3} - a_{2} = 1$
Then $a_{1} = a_{0}, a_{2} = \frac{a_{0}}{2}, a_{3} = \frac{2 + a_{0}}{6}, a_{4} = \frac{2 + a_{0}}{4!}, a_{5} = \frac{2 + a_{0}}{24 \cdot 4}$
Then the solution is $y = a_{0} + a_{0} x^{1} + \frac{a_{0}}{2} x^{2} + \frac{2 + a_{0}}{6} x^{3} + \frac{2 + a_{0}}{4!} x^{4} + \frac{2 + a_{0}}{24 \cdot 4} x^{5} \dots$
Is it correct? Am I supposed to find $a_{0}$ ?

Zack Wise 2022-04-23

Please solve this differential equation
$x + y y^{'} = 3 y^{2} y^{'} \sqrt{x^{2} + y^{2}}$

Travis Caldwell 2022-04-23

How to find the limit
$lim_{n \to \infty} \prod_{j = 1}^{n} (1 + \frac{1}{j} (\cos (\frac{t j}{n}) - 1))$

wuntsongo0cy 2022-04-23

How to find $lim_{n \to \infty} ({2 \sqrt{n}} - \sum_{k = 1}^{n} \frac{1}{\sqrt{k}})$ ?

Deven Livingston 2022-04-23

How to find first integral in a given quasilinear partial differential equation?
$y (x + y) \frac{\partial z}{\partial x} + x (x + y) \frac{\partial z}{\partial y} = 2 (x^{2} - y^{2}) + x z - y z .$

Elise Winters 2022-04-23

How many solutions does the ODE have?
Given:
${\begin{cases} y^{'} - a^{2} (y^{'})^{3} - \frac{\sin (x)}{x + y} = 0 \\ y (0) = 1 \end{cases}$
Write how many solutions does the system have for $a = 0$ and $a \neq 0$ .

Ali Marshall 2022-04-23

How does one sum the given series:
$\frac{1}{2 n + 1} + \frac{1}{2} \cdot \frac{1}{2 n + 3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{1}{2 n + 5} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{1}{2 n + 7} + \dots$

hadaasyj 2022-04-23

How to solve $X_{\times} = (1 + δ {(x)}) X$
I am trying to find the vibration modes of a string that has a uniform mass density, plus some point mass somewhere attached to it, modelled by an additional Dirac delta function in the mass density. The wave equation is of the form
$u_{x x} = (1 + δ (x)) u_{t t}$
where u is the deformation, and $(1 + δ (x))$ the mass density. After separation of variables we find
$X_{\times} = (1 + δ {(x)}) X$
where X is the spatial part of the solution. Is there any analytical solution for X?

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