What is a solution to the differential equation $y\frac{dy}{dx}={e}^{x}$ with y(0)=4?

ubumanzi18
2022-09-12
Answered

What is a solution to the differential equation $y\frac{dy}{dx}={e}^{x}$ with y(0)=4?

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asked 2022-09-12

What is the general solution of the differential equation $y\prime \prime +4y=0$?

asked 2022-05-21

I'm trying to solve this differential equation:

$\frac{dy}{dx}=\frac{x-\mathrm{exp}(y)}{y+\mathrm{exp}(y)}$

I thought I could use separation of variable, but I'm unable to isolate x.

Could you please help me get started on this differential equation?

$\frac{dy}{dx}=\frac{x-\mathrm{exp}(y)}{y+\mathrm{exp}(y)}$

I thought I could use separation of variable, but I'm unable to isolate x.

Could you please help me get started on this differential equation?

asked 2022-07-07

I am aware that the solution to a homogeneous first order differential equation of the form $\frac{dy}{dx}=p(x)y$ can be obtained by simply by rearranging to:

$\frac{dy}{y}=p(x)dx$

Then it is simply a question of integrating both sides and the answer is straightforward. Now what would happen if RHS had a constant, how a can find a particular solution to this case: $\frac{dy}{dx}=p(x)y+C$

I know that the general solution would be the sum of the homogeneous equation and the particular solution

$\frac{dy}{y}=p(x)dx$

Then it is simply a question of integrating both sides and the answer is straightforward. Now what would happen if RHS had a constant, how a can find a particular solution to this case: $\frac{dy}{dx}=p(x)y+C$

I know that the general solution would be the sum of the homogeneous equation and the particular solution

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 2022-01-21

Are the following are linear equation?

$1.\text{}\mathrm{sin}\left(x\right)\frac{dy}{dx}-3y=0$

$2.\text{}\frac{{d}^{2}y}{{dx}^{2}}+\mathrm{sin}\left(x\right)\frac{dy}{dx}=\mathrm{cos}\left(x\right)$

asked 2022-07-01

The differential equation that describes my system is given as

${y}^{(n)}+{a}_{n-1}{y}^{(n-1)}+\cdots +{a}_{1}\dot{y}+{a}_{0}y={b}_{n-1}{u}^{(n-1)}+\cdots +{b}_{1}\dot{u}+{b}_{0}u+g(y(t),u(t))$

I want to express the above differential equation into a system of linear differential equations of the form

$\dot{x}=Ax+Bu+{B}_{p}g$

$y=Cx$

The matrices are given as follows: However, I am not able to prove, how to get them

$A=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ -{a}_{0}& -{a}_{1}& -{a}_{2}& \cdots & -{a}_{n-1}\end{array}\right]$

$C=\left[\begin{array}{ccccc}1& {b}_{1}/{b}_{0}& {b}_{2}/{b}_{0}& \cdots & {b}_{n-1}/{b}_{0}\end{array}\right]$

How do I get the above matrices from the differential equation form as shown above?

${y}^{(n)}+{a}_{n-1}{y}^{(n-1)}+\cdots +{a}_{1}\dot{y}+{a}_{0}y={b}_{n-1}{u}^{(n-1)}+\cdots +{b}_{1}\dot{u}+{b}_{0}u+g(y(t),u(t))$

I want to express the above differential equation into a system of linear differential equations of the form

$\dot{x}=Ax+Bu+{B}_{p}g$

$y=Cx$

The matrices are given as follows: However, I am not able to prove, how to get them

$A=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ -{a}_{0}& -{a}_{1}& -{a}_{2}& \cdots & -{a}_{n-1}\end{array}\right]$

$C=\left[\begin{array}{ccccc}1& {b}_{1}/{b}_{0}& {b}_{2}/{b}_{0}& \cdots & {b}_{n-1}/{b}_{0}\end{array}\right]$

How do I get the above matrices from the differential equation form as shown above?

asked 2022-09-20

How do you solve the differential equation $y\prime ={e}^{-y}(2x-4)$, where y5)=0 ?