Detailed Explanations for Calculus 2 Equations

How to solve homogenous differential equation ${(x + y)}^{2} d x = x y d y$ ?
First i find value of $\frac{d y}{d x} = \frac{{(x + y)}^{2}}{x} y (i)$
let $y = v x$
differentiating both sides: $\frac{d y}{d x} = v + x d \frac{v}{d x} (i i)$
I tried to solve using equation (i) and (ii) but I am stuck.
${(x + y)}^{2} d x = x y d y$

Paula Boyer 2022-04-22

Particular solution of a system of second order ODE
$[\begin{matrix} - k & k \\ k & - k \end{matrix}] [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}] + [\begin{matrix} F - k \times a \\ k \times a \end{matrix}]$
$= \frac{d^{2}}{d t^{2}} [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}]$

Ali Marshall 2022-04-22

My textbook states we need locally Lipschitz continuous with respect to the second argument and uniformly with respect to the first argument of $f (t, x)$ . This is said to be equivalent to the result that for all compact subsets $V \subset U$ where $U \subseteq R^{n + 1}$ is open we have
$L = \supset_{(t, x) \neq (t, y) \in V \subset U} \frac{| f (t, x) - f (t, y) |}{| x - y |} < \infty$
I understand that locally Lipschitz means there is a neighborhood around each point where f is Lipschitz, but what does the uniform part mean with respect to t?

Sydney Stanley 2022-04-22

Instability of a parameter varying system whose parameters belong to a compact set:
Suppose, there is a system
$\dot{x} = f (t, γ_{p} (t), x)$
with $x \in R^{2}$ . For my specific case, parameter vector $γ_{p}$ is a scalar and known monotonic function with a compact image set (for example $γ_{p} (t) = e^{- t}$ , thus $γ_{p} \in [1, 0)$ ). I would like to show instability of this system

windpipe33u 2022-04-22

Solution to a system of ODE:s
Consider a system of linear ODE:s:
$x^{'} (t) = A x (t) + C$
Here A and C are coefficient matrices. If $C = 0$ , the solution is $x (t) = \exp (A x (t)) x (0)$ , where exp represents the matrix exponential. Can we write a similar solution for a general C?

Mekhi Cox 2022-04-22

Solution to
$\frac{1}{6} g^{″} = v g^{3} + k g^{3} - k g$

Judith Warner 2022-04-22

Sinusoids as solutions to differential equations
It is well known that the function
$t \mapsto a \cos (ω t) + b \sin (ω t)$
is the solution to the differential equation:
$x^{″} (t) = - ω^{2} x (t)$
with the initial conditions $x (0) = a and x^{'} (0) = b ω$ . I was wondering what differential equation will be solved by
$f (t) = \sum_{j = 1}^{k} (a_{j} \cos (ω_{j} t) + b_{j} \sin (ω_{j} t))$ .
It is obvious that this satisfies $x (t) = \sum_{j = 1}^{k} x_{j} (t)$ with $x_{j}^{″} (t) = - ω_{j}^{2} x_{j} (t)$ and initial conditions on $x_{j} (0)$ and $x_{j}^{^{'}} (0)$ . I was wondering if there were any other (perhaps more natural) differential equations that are satisfied by f.

dreangannaa 2022-04-22

Showing that continuous and differentiable s(t), with $s (0) = 0$ and $s^{'} (t) \leq 2 t s (t) + \sqrt{s (t)}$ for $t > 0$ , is identically zero for $t \geq 0$ .
Let $s : [0, + \infty) \to [0, + \infty)$ be a continuous function that is differentiable on $(0, + \infty)$ .
It also holds that $s^{'} (t) \leq 2 t s (t) + \sqrt{s (t)}, t > 0 s (0) = 0$
Show that $s (t) = 0, t \geq 0$ .
Do we us some theorem, maybe Rolle or the mean value theorem?
Do we have to check maybe the sign of $2 t s (t) + \sqrt{s (t)}$ ? This depends on the value of s(t), or not?

juniorychichoa70 2022-04-22

Poincare Map for Polar ODE
I am currently trying to obtain a Poincare Map for the ODE system originally given by
${\begin{array}{ll} \dot{x} = (1 - x^{2} - y^{2}) x - y \\ \dot{y} = x + (1 - x^{2} - y^{2}) y \end{array}$
on the region $x \in (\frac{1}{2}, \frac{3}{2})$ and $y = 0$ . Since $x^{2} + y^{2} = r^{2}$ and $\tan (θ) = \frac{y}{x}$ , we obtain that
${\begin{array}{ll} \dot{r} = \frac{x \dot{x} + y \dot{y}}{r} \\ \sec^{2} (θ) \dot{θ} = \frac{x \dot{y} - y \dot{x}}{x^{2}} \end{array} ⟹ {\begin{array}{ll} \dot{r} = r - r^{3} \\ \dot{θ} = 1 \end{array}$
However, I am stuck here with trying to identify the Poincare Map for the given system. Are there any recommendations for how to proceed? Moreover, how can I linearize this system at the point $(x, y) = (1, 0)$ (or in polar coordinates $(r, θ) = (1, 0)$ ?

Lymnmeatlypamgfm 2022-04-22

Possible Close-form solution for 2 dimensional $\frac{d Σ}{d t} = A Σ + Σ A^{T} + B B^{T}$
When A,B are 2x2 matrix and $\sum (t)$ are PSD, can we expect a close form solution for the following ODE,
$\frac{d Σ}{d t} = A Σ + Σ A^{T} + B B^{T} .$
The problem is from solving the covariance of a time-invariant SDE $x = A x + B d w .$
When x is in two dimensions, I guess there could be a close-form solution for $\sum$ . But I am not sure how to solve it.

hestyllvql 2022-04-22

How am I supposed to find the solution to the non-homogeneous ode ?
${\vec{Y}}^{'} = (\begin{matrix} - 4 & 2 \\ 2 & - 1 \end{matrix}) \vec{Y} + (\begin{matrix} x^{- 1} \\ 2 x^{- 1} + 4 \end{matrix})$

acceloq3c 2022-04-22

Value of differential equation at non existent points in the original function.
$y^{2} + y = x^{3}$

esbagoar7kh 2022-04-22

Solving
$y + x y^{'} = a (1 + x y)$
$y (\frac{1}{a}) = - a$

Esther Hoffman 2022-04-22

Solve linear system of ODE's
It's been years since I formally saw ODE's and I am quite rusty, I don;t remember how to solve linear ODE's.
I have a problem and managed to derive the following system:
${\begin{cases} 2 \ddot{q_{1}} = 2 q_{1} + q_{2} \\ 2 \ddot{q_{2}} = q_{1} \end{cases}$
where $q_{1} and q_{2}$ are scalar functions with parameter t.
It seems to me the solution is a trig function but I am not sure how to go about this problem.

Madilynn Avery 2022-04-21

Nonlinear first oder ODE
I want to solve the following nonlinear ODE
$y^{'} - \frac{y}{x} + \frac{3}{y} (\frac{1}{2} - x) + 1 = 0$ .

Ali Marshall 2022-04-21

Is any direct Adams Moulton formula or how to related those two formula which i am mention.
Are those formulas the same? When I am programming for Adams Moulton fourth order methods I use the 2nd formula which I mentioned in the 2nd image. Is it correct?
If correct, please explain me.
$Y_{n + 1} = Y_{n} + \frac{h}{24} [9 f (x_{n + 1}, y_{n + 1}) + 19 f (x_{n}, y_{n}) - 5 f (x_{n - 11} y_{n - 1}) + f (x_{n - 2}, y_{n - 2})$

Ari Sheppard 2022-04-21

If $a_{1}, \dots, a_{n}$ are in arithmetic progression and $a_{i} > 0$ for all i, then how to prove the following two identities:
1) $\frac{1}{\sqrt{a_{1}} + \sqrt{a_{2}}} + \frac{1}{\sqrt{a_{2}} + \sqrt{a_{3}}} + \dots + \frac{1}{\sqrt{a_{n - 1}} + \sqrt{a_{n}}} = \frac{n - 1}{\sqrt{a_{1}} + \sqrt{a_{n}}}$
2) $\frac{1}{a_{1} \cdot a_{n}} + \frac{1}{a_{2} \cdot a_{n - 1}} + \frac{1}{a_{3} \cdot a_{n - 2}} + \dots + \frac{1}{a_{n} \cdot a_{1}} = \frac{2}{a_{1} + a_{n}} (\frac{1}{a_{1}} + \frac{1}{a_{2}} + \dots + \frac{1}{a_{n}})$

Davon Trujillo 2022-04-21

Verify that the following function $y^{2} = e^{(2 x)} + C$ is a solution of the ODE: $y y^{'} = e^{(2 x)}$

Kymani Shepherd 2022-04-21

Verify $y = x \tan x$ is solution to ODE: $x y^{'} = y + x^{2} + y^{2}$
Verify $S y = x \tan {x}$
is a solution to $x y^{'} = y + x^{2} + y^{2}$

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