Detailed Explanations for Calculus 2 Equations

How to solve a complicated ODE
The equation is $f^{^{'}} (x) = γ \frac{f (x) + f^{2} (x)}{\log (\frac{f (x)}{1 + f (x)})}$
with the initial condition f(0), where $x \geq and f (0) \geq 0$ .
The solution is
$f (x) = \frac{1}{- 1 + \exp \sqrt{2 γ x + {\log (\frac{f (0) + 1}{f (0)})}^{2}}}$
I think the ODE can be solved by separating the variables
$\int_{0}^{f (x)} \frac{\log (\frac{f (z)}{1 + f (z)})}{f (z) + f^{2} (z)} d f = \int_{0}^{x} γ d z$
The right-hand size is easy. I do not know how to solve the integration of the left-hand side.

Zoie Phillips 2022-03-31

Second-order ODE involving two functions
I am wondering how to find a general analytical solution to the following ODE:
$\frac{d y}{d t} \frac{d^{2} x}{{d t}^{2}} = \frac{d x}{d t} \frac{d^{2} y}{{d t}^{2}}$
The solution method might be relatively simple; but right now I don't know how to approach this problem.

Caerswso1pc 2022-03-31

Second Order Nonhomogeneous Differential Equation (Method of Undetermined Coefficients)
Find the general solution of the following Differential equation $y^{″} - 2 y^{'} + 10 y = e^{x} \cos (3 x)$ . We know that the general solution for 2nd order Nonhomogeneous differential equations is the sum of $y_{p} + y_{c}$ where $y_{c}$ is the general solution of the homogeneous equation and $y_{p}$ the solution of the nonhomogeneous. Therefore $y_{c} = e^{x} (c_{1} \cos (3 x) + c_{2} \cos (3 x))$ . Now we have to find $y_{p}$ . I know in fact that $y_{c} = e^{x} (c_{1} \cos (3 x) + c_{2} \cos (3 x))$ . Now we have to find yp. I know in fact that $y_{p} = e^{\times} \frac{\sin (3 x)}{6}$ but i do not know how to get there.

Anahi Solomon 2022-03-31

Removal of absolute signs
An object is dropped from a cliff. The object leaves with zero speed, and t seconds later its speed v metres per second satisfies the differential equation
$\frac{d v}{d t} = 10 - 0.1 v^{2}$
So I found t in terms of v
$t = \frac{1}{2} \ln | \frac{10 + v}{10 - v} |$
The questions goes on like this: Find the speed of the object after 1 second. Part of the answer key shows this
$t = \frac{1}{2} \ln | \frac{10 + v}{10 - v} |$
$2 t = \ln | \frac{10 + v}{10 - v} |$
$e^{2 t} = \frac{10 + v}{10 - v}$
So here's my question: why is it not like this?
$\pm e^{2 t} = \frac{10 + v}{10 - v}$
Why can you ignore the absolute sign?

Octavio Chen 2022-03-30

I have encountered a question that goes like
$d \frac{d y}{d x} = y^{3} + x \cos (x), y (0) = 0 .$
Find the approximate solution which is $o (x^{5})$

Landin Harmon 2022-03-30

How to solve this equation by integration
$\frac{x^{″}}{x} - \frac{x^{'}}{x^{2}} - \frac{x^{' 2}}{x^{2}} = 0,$

atenienseec1p 2022-03-30

How to solve the differential equation
$y^{'} = 2 y (x \sqrt{y} - 1)$

metalskaashw 2022-03-30

How to solve 2nd order inhomogeneous differential equation ivp
$\ddot{y} + 4 y = t - 2 \sin (2 t), y (π) = 0, \dot{y} (π) = 1$

Jackson Floyd 2022-03-30

How to solve
$y^{″} + (1 - y) (1 + y) y = 0$

avalg10o 2022-03-30

$\frac{d y}{d x} = y + 1$ Solving the above given differential equation, yields the following general solution. $y + 1 = e^{x + C}$
$y = C e^{x} - 1$
$\Rightarrow$ Solution $y = - 1$ at $C = 0$
Can I say that $y = - 1$ is a singular solution?

calcolare45pj 2022-03-30

Proved that question:
$S (n) = \sum_{k \geq 1} \frac{n!}{k (n - k)! n^{k}} = \sum_{k \geq 1} \frac{n^{\underset{―}{k}}}{k n^{k}} \approx \frac{1}{2} \log (n)$

Averie Ferguson 2022-03-30

Second order differential inequality and comparison theorem.
If $x : [0, 1] \to R, x \in C^{\infty}$ that satisfies
${\begin{cases} \ddot{x} \leq - 2 x \\ x (0) = x (1) = 0 \end{cases}$
is it true that $x \geq 0 on [0, 1]$ ?
I observed that if $x \geq 0$ , then $\ddot{x} \leq 0$ , so x is concave. However I could not find out any other properties. I think if there exists a solution of the following differential equation
${\begin{cases} \ddot{x} = - 2 x \\ x (0) = x (1) = 0 \end{cases}$
then some kind of comparative theorem can be used. However, there is no non-trivial solution for this.
If the above proposition is incorrect, could you give me a counterexample?

Nathen Peterson 2022-03-30

Question in population dynamics using exponential growth rate equation
Given population doubles in 20 minutes, what is intrinsic growth rate r?
Attempt: Given population doubles, using exponential growth rate we have $\frac{d N}{d t} = 2 N$ so $N (t) = N_{0} e^{2 t}$ therefore $r = 2$ , but I have a feeling this is wrong since 20 minutes should be used somewhere around here.

Milton Robertson 2022-03-30

$x^{2} y^{″} + (3 x^{2} + 4 x) y^{'} + 2 (x^{2} + 3 x + 1) y = 0$
I try to solve this equation by finding $μ$ such that the equation become exact.( I know there are other ways for solving this equation).
${(μ y^{'})}^{'} = μ^{'} y^{'} + μ y^{″} = μ y^{″} + μ \frac{3 x^{2} + 4 x}{x^{2}} y^{'} + μ \frac{2 (x^{2} + 3 x + 1)}{x^{2}} y \Rightarrow$
$μ^{'} y^{'} = μ \frac{3 x^{2} + 4 x}{x^{2}} y^{'} + μ \frac{2 (x^{2} + 3 x + 1)}{x^{2}} y \Rightarrow \frac{μ^{'}}{μ} = \frac{3 x^{2} + 4 x}{x^{2}} + \frac{2 (x^{2} + 3 x + 1)}{x^{2}} \frac{y}{y^{'}}$
How am I supposed to solve it?

sexoagotadorogyr 2022-03-30

$y^{″} + 16 y = - \frac{2}{\sin (4 x)}$
I try to solve this ode using the variation of parameters theorem.
The characteristic polynomial of the homogenous equation is $r^{2} + 16 = 0$ .
Then $u_{1} (x) = \sin (4 x), u_{2} (x) = \cos (4 x)$
$y (x) = c_{1} (x) \sin (4 x) + c_{2} (x) \cos (4 x)$
I $c_{1}^{'} (x) \sin (4 x) + c_{2}^{'} \cos (4 x) = 0$
II $c_{1}^{'} (x) \cos (4 x) - c_{2}^{'} \sin (4 x) = - \frac{2}{\sin (4 x)}$
Multiply I by $\cos (4 x)$ and II by sin(4x) and subtract.
$c_{1}^{'} (\sin (4 x) \cos (4 x) - \cos (4 x) \sin (4 x)) + c_{2}^{'} (\cos^{2} (4 x) + \sin^{2} (4 x)) = 2$
We get $c_{2}^{'} = 2 \Rightarrow c_{2} = 2 x$ ,
$c_{1}^{'} = - \frac{2 \cos (4 x)}{\sin (4 x)} = - 2 \cot (4 x) \Rightarrow c_{1} = - \frac{1}{2} \ln | \sin (4 x) | .$
I don't get why it incorrect, where am I wrong?

Roy Roth 2022-03-30

Proof of Nagumo's Theorem of invariance
Given a continuous system $x^{'} (t) = p (x (t)), t \geq 0$ on $R^{n}$ with initial data $x (0) \in R^{n}$ and assuming that solutions exist and are unique in $R^{n}$ , let $S \subset R^{n}$ be a closed set. Then, S is positively invariant under the flow of the system $x (t) \in S$ for all $t \geq 0$ ) if and only if $p (x) \in K_{x} (S)$ for $x \in \partial S$ (the boundary of S) where $K_{x} (S)$ is the set of all sub-tangential vectors to S at x, i.e.,
$K_{x} (S) = {z \in R^{n} : lim_{h - \to 0} d \frac{d (x + h p (x); S) = 0}{h}},$
where d(w;S) is the distance from the vector w to the subset S.

Nishil Savla2022-03-29

Show that f(x)=1/x does not have extreme values in interval (0,1).

Oliver Tyler 2022-03-29

I have to solve the following Cauchy's problem:
${\begin{aligned} x^{2} x^{'} = \sin^{2} (x^{3} - 3 t) \\ x (0) = 1 \end{aligned}$

palmantkf4u 2022-03-29

How would we solve the same initial value problem but instead of $c \in R$ being a constant we have a function $f : R^{n} \times [0, t) \to R$ in its place such that the problem becomes:
${\begin{cases} u_{t} + b \cdot (D_{x} u) + f (x, t) u = h (x, t) & in R^{n} \times (0, \infty) \\ u (x, 0) = g (x) & on R^{n} \times {t = 0} \end{cases}$
What steps would I take to find a function u(x,t) that satisfies this? I am assuming u(x,t) will be similar to the function I found for the original problem but with some additional integrals (after playing with it for a little), but I am unsure.
I am fairly new to partial differential equations so any help will be appreciated.

Octavio Chen 2022-03-29

How to solve this diffrential equation using parts formula?
$\frac{d y}{d x} + \frac{y}{20} = 50 (1 + \cos x)$

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