# Differential equations questions and answers

Recent questions in Differential Equations

### I was wondering if I could get some advice on how to tackle this question:Consider the differential equation${x}^{2}\frac{dy}{dx}+2xy-{y}^{3}=0\phantom{\rule{1em}{0ex}}\left(3\right)$Make the substitution $u={y}^{-2}$ and show that the differential equation reduces to$-\frac{1}{2}{x}^{2}\frac{du}{dx}+2xu-1=0\phantom{\rule{1em}{0ex}}\left(4\right)$Solve equation (4) for u(x) and hence write down the solution for equation (3).I'm trying to do the first part of showing that the differential equation reduces to equation 4. I have started out by:$\begin{array}{rl}u& ={y}^{-2}\\ & =\frac{1}{{y}^{2}}\\ \therefore {y}^{2}& =\frac{1}{u}\\ \phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}y& =±\sqrt{\frac{1}{u}}\end{array}$I'm not sure where to continue on from here though.

Spencer Lutz 2022-05-15 Answered

### I am given this:$\left(2x+1\right)\frac{dy}{dx}+y=0$I tried this:$\frac{1}{\left(2x+1\right)}dx=\frac{-1}{y}dy$Then integrated the above sum and got this:$\frac{ln\left(2x+1\right)}{2}=-ln\left(y\right)$The answer is: ${y}^{2}\left(2x+1\right)=C$.I tried solving it by placing the like terms together and integrating them. However, my answer is wrong from the answer given. Could you point out my mistake? Or am i evaluating the entire thing incorrectly?All suggestions and help are appreciated!

Jaylene Duarte 2022-05-13 Answered

### I've been trying to solve the following differential equation:$3x+y\left(x\right)-2+\frac{dy\left(x\right)}{dx}\left(x-1\right)=0$And I found out it's an exact differential equation, since it can be rearranged as $\left(3x+y\left(x\right)-2\right)dx+\left(x-1\right)dy=0$Assuming that a function $U\left(x,y\right)=\frac{\mathrm{\partial }U}{\mathrm{\partial }x}dx+\frac{\mathrm{\partial }U}{\mathrm{\partial }y}dy=k$, (where $k\equiv$ constant) exists, I calculated it as:$U\left(x,y\right)=\int \left(3x+y-2\right)dx+\int \left(x-1\right)dy=k$And I got$y\left(x\right)=\frac{-3{x}^{2}}{2\left(2x-1\right)}+\frac{2x}{2x-1}+\frac{k}{2x-1}$But Mathematica says the solution is$y\left(x\right)=\frac{-3{x}^{2}}{2\left(x-1\right)}+\frac{2x}{x-1}+\frac{k}{1-x}$So either I assumed something which isn't correct or I made a mistake along the process. Where did I go wrong?

Elle Weber 2022-05-13 Answered

### Let $f$ be a differentiable function such that $f\left(3\right)=2.345$ and ${f}^{\prime }\left(x\right)=ln\left({x}^{2}+1\right)$. What is the value of $f\left(5\right)$?

London Ware 2022-05-10 Answered

### Linear approximation to find $\frac{1}{4.002}$

Carley Haley 2022-05-10 Answered

### Use linear approximation to approximate $1/0.254$.

Angelique Horne 2022-05-09 Answered

### Use linear approximation, i.e. the tangent line, to approximate ${11.2}^{2}$ as follows :Let $f\left(x\right)={x}^{2}$ and find the equation of the tangent line to $f\left(x\right)$ at $x=11$. Using this, find your approximation for ${11.2}^{2}$.

Alexis Meyer 2022-05-09 Answered

### Use the linear approximation of $f\left(x,y,z\right)=$ at $\left(-2,1,1\right)$ to estimate $f\left(-1.98,0.97,1.03\right)$

Paul Duran 2022-05-08 Answered

### Using $f\left(x,y,z\right)=\sqrt{x+2y+3z}$ and an appropriately chosen linear approximation to find the value of $\sqrt{226}$

Speaking of differential equations, these are used not only by those students majoring in Physics because solving differential equations is also quite common in Statistics and Financial Studies. Explore the list of questions and examples of equations to get a basic idea of how it is done.

These answers below are meant to provide you with the starting points as you work with your differential equations. If you need specific help or cannot understand the rules behind the answers that are presented below, start with a simple equation and learn with the provided solutions..