Solve the first order differential equation using any acceptable method.

$\mathrm{sin}\left(x\right)\frac{dy}{dx}+\left(\mathrm{cos}\left(x\right)\right)y=0$ , $y\left(\frac{7\pi}{6}\right)=-2$

tebollahb
2022-01-20
Answered

Solve the first order differential equation using any acceptable method.

$\mathrm{sin}\left(x\right)\frac{dy}{dx}+\left(\mathrm{cos}\left(x\right)\right)y=0$ , $y\left(\frac{7\pi}{6}\right)=-2$

You can still ask an expert for help

asked 2022-06-23

Given a second order differential equation

${y}^{\u2033}+f(y){y}^{\prime}+g(y)=0,$

write an equivalent system of first order equations with transformations

${x}_{1}=y,{x}_{2}={y}^{\prime}+{\int}_{0}^{y}f(s)ds.$

This is what I did:

${x}_{1}^{\prime}={y}^{\prime}={x}_{2}-{\int}_{0}^{y}f(s)ds={x}_{2}-{\int}_{0}^{{x}_{1}}f(s)ds$

${x}_{2}^{\prime}={y}^{\u2033}+f(y)=-f(y){y}^{\prime}-g(y)+f(y)=f(y)(1-{y}^{\prime})-g(y)=f({x}_{1})(1-{x}_{2}+{\int}_{0}^{{x}_{1}}f(s)ds)-g({x}_{1})$

I feel like this answer is wrong though, because I am not sure if I'm doing the standard procedure.

${y}^{\u2033}+f(y){y}^{\prime}+g(y)=0,$

write an equivalent system of first order equations with transformations

${x}_{1}=y,{x}_{2}={y}^{\prime}+{\int}_{0}^{y}f(s)ds.$

This is what I did:

${x}_{1}^{\prime}={y}^{\prime}={x}_{2}-{\int}_{0}^{y}f(s)ds={x}_{2}-{\int}_{0}^{{x}_{1}}f(s)ds$

${x}_{2}^{\prime}={y}^{\u2033}+f(y)=-f(y){y}^{\prime}-g(y)+f(y)=f(y)(1-{y}^{\prime})-g(y)=f({x}_{1})(1-{x}_{2}+{\int}_{0}^{{x}_{1}}f(s)ds)-g({x}_{1})$

I feel like this answer is wrong though, because I am not sure if I'm doing the standard procedure.

asked 2022-01-21

Help to solve the following first order differential equations:

a.$x{y}^{4}dx+({y}^{2}+2){e}^{-5x}dy=0$

b.$(x+1){y}^{\prime}=x+6$

a.

b.

asked 2022-05-20

Recently I have met such an equation:

$x\frac{dy}{dx}=y+\sqrt{{x}^{2}+{y}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{dy}{dx}-\frac{y}{x}=\frac{\sqrt{{x}^{2}+{y}^{2}}}{x}$

First of all what type it has? I can not refer it to any I am aware of(I am a newbie so it is likely that I'm missing something). Trying to solve it gave no result therefore I googled and youtube revealed that I had to use change of variables $y=v(x)x$. However it looks like a cheating. Why use this particular replacement and is there any way to solve it without it?

$x\frac{dy}{dx}=y+\sqrt{{x}^{2}+{y}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{dy}{dx}-\frac{y}{x}=\frac{\sqrt{{x}^{2}+{y}^{2}}}{x}$

First of all what type it has? I can not refer it to any I am aware of(I am a newbie so it is likely that I'm missing something). Trying to solve it gave no result therefore I googled and youtube revealed that I had to use change of variables $y=v(x)x$. However it looks like a cheating. Why use this particular replacement and is there any way to solve it without it?

asked 2022-09-20

How do I solve the nonlinear differential equation $y\prime =-\frac{x}{y}$ under initial condition y(1)=1?

asked 2022-06-16

Given non-commuting matrices $A$ and $B$ of order $n$, is there a closed-form solution to the differential equation

$\frac{dX}{dt}=AX+tBX$

with $X(0)=I$?

I know that for the reals, $x=a\mathrm{exp}\int f(t)$ is the general solution to $\dot{x}=f(t)x$, but I'm also 99% certain this relies on the commutivity of the reals.

I'm more specifically looking to numerically compute $X(T)$ given the more general differential equation

$\frac{dX}{dt}=f(t)X,\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}X(0)=I$

but in circumstances where ${f}^{\prime}(t)$ may be large and a 1st order piecewise approximation would be far more accurate than 0th order for any given $\mathrm{\Delta}t$. Ultimately my concern is computing $X(T)$ as quickly as possible.

Are there better techniques for accomplishing this?

$\frac{dX}{dt}=AX+tBX$

with $X(0)=I$?

I know that for the reals, $x=a\mathrm{exp}\int f(t)$ is the general solution to $\dot{x}=f(t)x$, but I'm also 99% certain this relies on the commutivity of the reals.

I'm more specifically looking to numerically compute $X(T)$ given the more general differential equation

$\frac{dX}{dt}=f(t)X,\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}X(0)=I$

but in circumstances where ${f}^{\prime}(t)$ may be large and a 1st order piecewise approximation would be far more accurate than 0th order for any given $\mathrm{\Delta}t$. Ultimately my concern is computing $X(T)$ as quickly as possible.

Are there better techniques for accomplishing this?

asked 2022-09-25

Solve the Differential Equation $\frac{dy}{dx}+3y=0$ with x=0 when y=1?

asked 2022-01-21

Classify the following system of first-order partial differential equations (k=const.):

$u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial u}{\partial x}+w\frac{\partial v}{\partial y}=0$

$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+k\frac{\partial w}{\partial y}=0$

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+k\frac{\partial w}{\partial x}=0$