 # Calculus 2 questions and answers

Recent questions in Calculus 2 iyiswad9k 2022-05-01 Answered

### Proving single solution to an initial value problemor each $\left({x}_{0},{y}_{0}\right)\in \mathbb{R}×\mathbb{R}$ I need to prove that there is a single solution defined on $\mathbb{R}$ Dashawn Robbins 2022-05-01 Answered

### Proving identity$\sum \left[{\left(j+t\right)}^{-1}-{j}^{-1}\right]=\sum _{k\ge 1}\zeta \left(k+1\right){\left(-t\right)}^{k}$ Caitlyn Cole 2022-04-30 Answered

### Suppose the function $f:\mathbb{R}\to \mathbb{R}$ is continuous. For a natural number $k$, let ${x}_{1},{x}_{2},...,{x}_{k}$ be points in $\left[a,b\right]$. Prove there is a point $z$ in $\left[a,b\right]$ at which$f\left(z\right)=\left(f\left({x}_{1}\right)+f\left({x}_{2}\right)+...+f\left({x}_{k}\right)\right)/k$So I'm thinking about applying the intermediate value theorem:If$a<{x}_{1}then$f\left(a\right)or$k.f\left(a\right)$f\left(a\right)<\left(f\left({x}_{1}\right)+f\left({x}_{2}\right)+...+f\left({x}_{k}\right)\right)/kBut I couldn't think of any way to prove that $f\left(a\right) or is it even true?EDIT: Thanks everyone for your effort. iyiswad9k 2022-04-30 Answered

### Let $I=\left[a,b\right]$ with $a and let $u:I\to \mathbb{R}$ be a function with bounded pointwise variation, i.e.$Va{r}_{I}u=sup\left\{\sum _{i=1}^{n}|u\left({x}_{i}\right)-u\left({x}_{i-1}\right)|\right\}<\mathrm{\infty }$where the supremum is taken over all partition $P=\left\{a={x}_{0}<{x}_{1}<...<{x}_{n-1}. How can I prove that if $u$ satisfies the intermediate value theorem (IVT), then $u$ is continuous?My try: $u$ can be written as a difference of two increasing functions ${f}_{1},{f}_{2}$. I know that a increasing function that satisfies the (ITV) is continuous, hence, if I prove that ${f}_{1},{f}_{2}$ satisfies the (ITV) the assertion follows. But, is this true? I mean, ${f}_{1},{f}_{2}$ satisfies (ITV)? adiadas8o7 2022-04-30 Answered

### Second-order ODE involving two functionsI am wondering how to find a general analytical solution to the following ODE:$\frac{dy}{dt}\frac{{d}^{2}x}{{dt}^{2}}=\frac{dx}{dt}\frac{{d}^{2}y}{{dt}^{2}}$The solution method might be relatively simple; but right now I don't know how to approach this problem. Kaiya Hardin 2022-04-30 Answered

### Second Order Nonhomogeneous Differential Equation (Method of Undetermined Coefficients)Find the general solution of the following Differential equation $y{}^{″}-2{y}^{\prime }+10y={e}^{x}\mathrm{cos}\left(3x\right)$. 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 ${y}_{c}={e}^{x}\left({c}_{1}\mathrm{cos}\left(3x\right)+{c}_{2}\mathrm{cos}\left(3x\right)\right)$. Now we have to find ${y}_{p}$. I know in fact that ${y}_{c}={e}^{x}\left({c}_{1}\mathrm{cos}\left(3x\right)+{c}_{2}\mathrm{cos}\left(3x\right)\right)$. Now we have to find yp. I know in fact that ${y}_{p}={e}^{×}\frac{\mathrm{sin}\left(3x\right)}{6}$ but i do not know how to get there. micelije1mw 2022-04-30 Answered

### Removal of absolute signsAn 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$\frac{dv}{dt}=10-0.1{v}^{2}$So I found t in terms of v$t=\frac{1}{2}\mathrm{ln}|\frac{10+v}{10-v}|$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}\mathrm{ln}|\frac{10+v}{10-v}|$$2t=\mathrm{ln}|\frac{10+v}{10-v}|$${e}^{2t}=\frac{10+v}{10-v}$So here's my question: why is it not like this?$±{e}^{2t}=\frac{10+v}{10-v}$Why can you ignore the absolute sign? Magdalena Norton 2022-04-30 Answered

### Question on a differential equationLet $x:\mathbb{R}⇒\mathbb{R}$ a solution of the differenzial equation:Proof that the function:$f:\mathrm{ℝ}⇒\mathrm{ℝ}$ Maurice Maldonado 2022-04-30 Answered

### How to solve this nonlinear diff eq of celestial mechanics?${\left(\stackrel{˙}{r}\right)}^{2}=\frac{2\mu }{r}+2h$Where mu and h are constants.I have no idea how to solve it, maybe there is a trick I didn't know.The only thing that came in mind is to integrate$\int \frac{dr}{\sqrt{\frac{2\mu }{r}+2h}}=\int dt$but I don't think this is really a solution, since I don't know too how to evaluate the integral in terms of elementary functions. Bailee Ortiz 2022-04-29 Answered

### Let's say for example these two:$\frac{3n}{2n+3}{\left(-1\right)}^{n+1}-\frac{{\left(-1\right)}^{n}}{2n}{\left({\left(-1\right)}^{n}+\frac{1}{n}\right)}^{2}$ Elise Winters 2022-04-29 Answered

### Let u be a solution of the differential equation ${y}^{\prime }+xy=0$ and $\varphi =u\psi$ be a solution of the differential equation ${y}^{″}+2x{y}^{\prime }+\left({x}^{2}+2\right)y=0$ such that $\varphi \left(0\right)=1,{\varphi }^{\prime }\left(0\right)=0$. Then $\varphi \left(x\right)$ is equal toA) $\left({\mathrm{cos}}^{2}\left(x\right)\right){e}^{\frac{-{x}^{2}}{2}}$B) $\left(\mathrm{cos}\left(x\right)\right){e}^{\frac{-{x}^{2}}{2}}$C) $\left(1+{x}^{2}\right){e}^{\frac{-{x}^{2}}{2}}$D) $\left(\mathrm{cos}\left(x\right)\right){e}^{-{x}^{2}}$ Alejandro Atkins 2022-04-29 Answered

### Let $f:\mathbb{R}⇒\mathbb{R}$ be a differentiable function, and suppose $f={f}^{\prime }$ and $f\left(0\right)=1$. Then prove $f\left(x\right)\ne 0$ for all $x\in \mathbb{R}$ wuntsongo0cy 2022-04-29 Answered

### Let N be a positive integer. Find all real numbers a such that the differential equation $\frac{{d}^{2}y}{d{x}^{2}}-4a\frac{dy}{dx}+3y=0$ has a nontrivial solution satisfying the conditions $y\left(0\right)=0$ and $y\left(2N\pi \right)=0$ nrgiizr0ib6 2022-04-29 Answered

### I have to solve this recurrence using substitutions:$\left(n+1\right)\left(n-2\right){a}_{n}=n\left({n}^{2}-n-1\right){a}_{n-1}-{\left(n-1\right)}^{3}{a}_{n-2}$ with ${a}_{2}={a}_{3}=1$ g2esebyy7 2022-04-29 Answered

### I have the differential equation$\frac{d}{dx}\left(p\left(x\right)\frac{df}{dx}\right)+\cdots =0$and I want to perform a generic change of variable from x to $y=y\left(x\right)$. Molecca89g 2022-04-29 Answered

### I have encountered a question that goes likeFind the approximate solution which is $o\left({x}^{5}\right)$ Judith Warner 2022-04-29 Answered

### How would we solve the same initial value problem but instead of $c\in \mathbb{R}$ being a constant we have a function $f:{\mathbb{R}}^{n}×\left[0,t\right)\to \mathbb{R}$ in its place such that the problem becomes: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. Wyatt Flores 2022-04-29 Answered

### How to you find the general solution of $\frac{dy}{dx}=x{\mathrm{cos}x}^{2}$? Lymnmeatlypamgfm 2022-04-29 Answered

### How to solve this ordinary differential equation$tx{}^{‴}+3x{}^{″}-t{x}^{\prime }-x=0$?For equation $tx{}^{‴}+3x{}^{″}-t{x}^{\prime }-x=0$, we know a special solution ${x}_{1}=\frac{1}{t}$, how to general solution?I firstly attempted $d\left(tx{}^{″}+2{x}^{\prime }-tx\right)=0$, then $tx2{x}^{\prime }-tx=C$, C is a constant.But in next step , I found that my solution is wrong.Since ${x}_{1}=\frac{1}{t}$ is a special solution of $tx{}^{‴}+3x{}^{″}-t{x}^{\prime }-x=0$, we found ${x}_{2}=-\frac{1}{t}$ is a solution of equation.Then $x={x}_{1}-{x}_{2}=\frac{2}{t}$ is a solution of $tx{}^{″}+2{x}^{\prime }-tx=0$. As you can see ,the step is wrong.Then I attemped other way to solve this equation ,but all failed.Could help me solve this equation? Maximillian Patterson 2022-04-29 Answered

### $y{}^{″}+16y=-\frac{2}{\mathrm{sin}\left(4x\right)}$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}\left(x\right)=\mathrm{sin}\left(4x\right),{u}_{2}\left(x\right)=\mathrm{cos}\left(4x\right)$$y\left(x\right)={c}_{1}\left(x\right)\mathrm{sin}\left(4x\right)+{c}_{2}\left(x\right)\mathrm{cos}\left(4x\right)$I ${c}_{1}^{\prime }\left(x\right)\mathrm{sin}\left(4x\right)+{c}_{2}^{\prime }\mathrm{cos}\left(4x\right)=0$II ${c}_{1}^{\prime }\left(x\right)\mathrm{cos}\left(4x\right)-{c}_{2}^{\prime }\mathrm{sin}\left(4x\right)=-\frac{2}{\mathrm{sin}\left(4x\right)}$Multiply I by $\mathrm{cos}\left(4x\right)$ and II by sin(4x) and subtract.${c}_{1}^{\prime }\left(\mathrm{sin}\left(4x\right)\mathrm{cos}\left(4x\right)-\mathrm{cos}\left(4x\right)\mathrm{sin}\left(4x\right)\right)+{c}_{2}^{\prime }\left({\mathrm{cos}}^{2}\left(4x\right)+{\mathrm{sin}}^{2}\left(4x\right)\right)=2$We get ${c}_{2}^{\prime }=2⇒{c}_{2}=2x$,${c}_{1}^{\prime }=-\frac{2\mathrm{cos}\left(4x\right)}{\mathrm{sin}\left(4x\right)}=-2\mathrm{cot}\left(4x\right)⇒{c}_{1}=-\frac{1}{2}\mathrm{ln}|\mathrm{sin}\left(4x\right)|.$I don't get why it incorrect, where am I wrong?

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.