 # Calculus 2 questions and answers

Recent questions in Calculus 2 hadaasyj 2022-04-28 Answered

### Proof of Nagumo's Theorem of invarianceGiven a continuous system ${x}^{\prime }\left(t\right)=p\left(x\left(t\right)\right),t\ge 0$ on ${\mathbb{R}}^{n}$ with initial data $x\left(0\right)\in {\mathbb{R}}^{n}$ and assuming that solutions exist and are unique in ${\mathbb{R}}^{n}$, let $S\subset {\mathbb{R}}^{n}$ be a closed set. Then, S is positively invariant under the flow of the system $x\left(t\right)\in S$ for all $t\ge 0$) if and only if $p\left(x\right)\in {K}_{x}\left(S\right)$ for $x\in \partial S$ (the boundary of S) where ${K}_{x}\left(S\right)$ is the set of all sub-tangential vectors to S at x, i.e.,${K}_{x}\left(S\right)\phantom{\rule{0.222em}{0ex}}=\left\{z\in {\mathbb{R}}^{n}:\underset{h-\to 0}{lim}\frac{d\left(x+hp\left(x\right);S\right)=0}{h}\right\},$where d(w;S) is the distance from the vector w to the subset S. luminarc24lry 2022-04-28 Answered

### Limit of a separable equation:Consider the equation$\frac{dy}{dt}=k\left(a-y\right)\left(b-y\right)$where a,b and k are constants. Assuming $y\left(0\right)=0$a) Solve for y(t) when $a=b$b) Solve for the case $0c) By considering the limit $b⇒a$ in (b) show that the two results are consistent Beedgighref28n 2022-04-28 Answered

### Limit for $\to \mathrm{\infty }$ of a solution to an ODEConsider the Cauchy problem: Sydney Stanley 2022-04-28 Answered

### Solve this nonhomegenous ode$y{}^{″}+4y=\mathrm{cos}\left(2x\right)$ Davin Sheppard 2022-04-28 Answered

### Solve the heat equation using a transform method$k\frac{{\partial }^{2}U}{\partial {x}^{2}}=\frac{\partial U}{\partial t}$subject to$U\left(0,t\right)=1,t>0$$U\left(x,0\right)={e}^{-x},x>0$ Essence Byrd 2022-04-28 Answered

### Solve the following differential equation by the form of homogeneous equation. Letting $y=vx$The equation: ${x}^{2}\frac{dy}{dx}+xy+1=0$ Dania Robbins 2022-04-28 Answered

### I am unable to solve the differential equation of first order given below,$\left(\mathrm{sin}y\mathrm{cos}y+x{\mathrm{cos}}^{2}y\right)dx+xdy=0$ Maurice Maldonado 2022-04-28 Answered

### How to solve this equation by integration$\frac{{x}^{″}}{x}-\frac{{x}^{\prime }}{{x}^{2}}-\frac{{x}^{\prime 2}}{{x}^{2}}=0,$ coraletsmmh 2022-04-28 Answered

### How to solve this diffrential equation using parts formula?$\frac{dy}{dx}+\frac{y}{20}=50\left(1+\mathrm{cos}x\right)$ Molecca89g 2022-04-27 Answered

### Let $f\in {C}_{o}^{\mathrm{\infty }}\left({\mathbb{R}}^{n}\right)$. Propose formulas of the form to solve:i) $\sum _{j=1}^{n}\frac{{\partial }^{4}u\left(x\right)}{\partial {x}_{j}^{4}}+u\left(x\right)=f\left(x\right)$andii) Ali Marshall 2022-04-27 Answered

### I have to solve the following Cauchy's problem:$\left\{\begin{array}{rl}& {x}^{2}{x}^{\prime }={\mathrm{sin}}^{2}\left({x}^{3}-3t\right)\\ & x\left(0\right)=1\end{array}.$ Zack Wise 2022-04-27 Answered

### I have two problems for which I know the answers (and working) but am still confused about the method used to solve them. The equations are In my notes it instructs that in the case of the independent variable missing from the equation (x and t, respectively), the substitution to reduce the order is made as follows:$p={y}^{\left(1\right)}$${y}^{\left(2\right)}=\frac{dp}{dx}=\frac{dp}{dy}\frac{dy}{dx}=p\frac{dp}{dy}$Then, for the first equation:$y{y}^{\left(2\right)}={\left({y}^{\left(1\right)}\right)}^{2}⇒yp\frac{dp}{dy}={p}^{2}$But for the second equation, the substitution is made as:${x}^{\left(2\right)}+{\left({x}^{\left(1\right)}\right)}^{2}=0⇒\frac{dp}{dt}=-{p}^{2}$I've tested this with online calculators and even they compute these two differential equations in the two different ways.Any explanation of where I'm going wrong would be greatly appreciated. luminarc24lry 2022-04-27 Answered

### Solve the series:$\underset{n\to \mathrm{\infty }}{lim}\frac{k+1}{{3}^{k}}$ kabutjv7 2022-04-27 Answered

### Solve the equation:$x\left(2y+1\right)dx=y\left({x}^{2}-3x+2\right)dy$ Poemslore8ye 2022-04-27 Answered

### I am trying to find the common ratio of $\sum _{n=0}^{\mathrm{\infty }}{2}^{-n}{z}^{{n}^{2}}$ Kale Mcclain 2022-04-27 Answered

### Solve the differential equation:$\frac{d}{dx}\left(2y{y}^{\prime }\right)={\left({y}^{\prime }\right)}^{2}$ dolovatgyp 2022-04-27 Answered

### I am interested in the following differential equation: $y{}^{″}-qy$ where $q:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}^{\cdot }$ is a continuous, positive function. Terrence Moore 2022-04-27 Answered

### Solve ${x}^{3}y\text{'}\text{'}\text{'}+xy\text{'}-y=x\mathrm{ln}\left(x\right)$using shift $x={e}^{z}$ and differential operator .What does  mean?$\left({e}^{z}{\right)}^{3}y\text{'}\text{'}\text{'}+{e}^{z}y\text{'}-y={e}^{z}\mathrm{ln}\left({e}^{z}\right)$$\left({e}^{3z}\right)y\text{'}\text{'}\text{'}+{e}^{z}y\text{'}-y={e}^{z}z$$y={z}^{r}$${e}^{3}r\left({r}^{2}-r-2\right){z}^{r-3}+{e}^{z}r{z}^{r-1}-{z}^{r}=0$Continue ? Dashawn Robbins 2022-04-27 Answered

### How to solve the ODE$y\text{'}=\frac{x-{e}^{x}}{x+{e}^{y}}$ misangela4gi 2022-04-27 Answered

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

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.