Patatiniuh

2022-07-12

I want to solve the following first order linear system of differential equations:

$\begin{array}{rl}& \frac{d{y}_{0}}{dx}+x{y}_{1}=\lambda {y}_{2}\\ & \frac{d{y}_{1}}{dx}+x{y}_{2}=\lambda {y}_{0}\\ & \frac{d{y}_{2}}{dx}+\frac{\alpha}{x}{y}_{2}+x{y}_{0}=\lambda {y}_{1}\end{array}$

with intial conditions: ${y}_{0}(0)=1,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{1}(0)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{2}(0)=0.$

$\begin{array}{rl}& \frac{d{y}_{0}}{dx}+x{y}_{1}=\lambda {y}_{2}\\ & \frac{d{y}_{1}}{dx}+x{y}_{2}=\lambda {y}_{0}\\ & \frac{d{y}_{2}}{dx}+\frac{\alpha}{x}{y}_{2}+x{y}_{0}=\lambda {y}_{1}\end{array}$

with intial conditions: ${y}_{0}(0)=1,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{1}(0)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{2}(0)=0.$

Nathen Austin

Beginner2022-07-13Added 14 answers

After putting your system into the form $\dot{\mathit{y}}(x)=\mathit{A}(x)\mathit{y}(x)$ you will see that the system is a linear time variant (LTV) differential equation. The system matrix is given by

$\mathit{A}(x)=\left[\begin{array}{ccc}0& -x& \lambda \\ \lambda & 0& -x\\ -x& \lambda & -\alpha /x\end{array}\right]$

your state vector is $\mathit{y}=[{y}_{0},{y}_{1},{y}_{2}{]}^{T}$.

Solving this system is not trivial. I gave this ODE to Maple but it was not able to solve it. You could also try Mathematica/Sympy/Mupad or other computer algebra systems.

There is a symbolic solution to LTV systems referred as the Peano-Baker series. In some cases, you might be able to find a closed form solution from this series, but it is not very likely for your problem.

If you just want to study the stability of some solutions then you might also consider using Floquet-Theory.

$\mathit{A}(x)=\left[\begin{array}{ccc}0& -x& \lambda \\ \lambda & 0& -x\\ -x& \lambda & -\alpha /x\end{array}\right]$

your state vector is $\mathit{y}=[{y}_{0},{y}_{1},{y}_{2}{]}^{T}$.

Solving this system is not trivial. I gave this ODE to Maple but it was not able to solve it. You could also try Mathematica/Sympy/Mupad or other computer algebra systems.

There is a symbolic solution to LTV systems referred as the Peano-Baker series. In some cases, you might be able to find a closed form solution from this series, but it is not very likely for your problem.

If you just want to study the stability of some solutions then you might also consider using Floquet-Theory.

- A body falls from rest against resistance proportional to the square root of the speed at any instant. If the body attains speed V1 and V2 feet per second, after 1 and 2 seconds in motion, respectively, find an expression for the limiting velocity.

The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True or False

The Laplace transform of

is$u(t-2)$

(a)$\frac{1}{s}+2$

(b)$\frac{1}{s}-2$

(c) ??$e}^{2}\frac{s}{s}\left(d\right)\frac{{e}^{-2s}}{s$ 1 degree on celsius scale is equal to

A) $\frac{9}{5}$ degree on fahrenheit scale

B) $\frac{5}{9}$ degree on fahrenheit scale

C) 1 degree on fahrenheit scale

D) 5 degree on fahrenheit scaleThe Laplace transform of $t{e}^{t}$ is A. $\frac{s}{(s+1{)}^{2}}$ B. $\frac{1}{(s-1{)}^{2}}$ C. $\frac{s}{(s+1{)}^{2}}$ D. $\frac{s}{(s-1)}$

What is the Laplace transform of

into the s domain?$t\mathrm{cos}t$ Find the general solution of the given differential equation:

${y}^{\u2033}-2{y}^{\prime}+y=0$The rate at which a body cools is proportional to the difference in

temperature between the body and its surroundings. If a body in air

at 0℃ will cool from 200℃ 𝑡𝑜 100℃ in 40 minutes, how many more

minutes will it take the body to cool from 100℃ 𝑡𝑜 50℃ ?A body falls from rest against a resistance proportional to the velocity at any instant. If the limiting velocity is 60fps and the body attains half that velocity in 1 second, find the initial velocity.

What's the correct way to go about computing the Inverse Laplace transform of this?

$\frac{-2s+1}{({s}^{2}+2s+5)}$

I Completed the square on the bottom but what do you do now?

$\frac{-2s+1}{(s+1{)}^{2}+4}$How to find inverse Laplace transform of the following function?

$X(s)=\frac{s}{{s}^{4}+1}$

I tried to use the definition: $f(t)={\mathcal{L}}^{-1}\{F(s)\}=\frac{1}{2\pi i}\underset{T\to \mathrm{\infty}}{lim}{\int}_{\gamma -iT}^{\gamma +iT}{e}^{st}F(s)\phantom{\rule{thinmathspace}{0ex}}ds$or the partial fraction expansion but I have not achieved results.How do i find the lapalace transorm of this intergral using the convolution theorem? ${\int}_{0}^{t}{e}^{-x}\mathrm{cos}x\phantom{\rule{thinmathspace}{0ex}}dx$

How can I solve this differential equation? : $xydx-({x}^{2}+1)dy=0$

Find the inverse Laplace transform of $\frac{{s}^{2}-4s-4}{{s}^{4}+8{s}^{2}+16}$

inverse laplace transform - with symbolic variables:

$F(s)=\frac{2{s}^{2}+(a-6b)s+{a}^{2}-4ab}{({s}^{2}-{a}^{2})(s-2b)}$

My steps:

$F(s)=\frac{2{s}^{2}+(a-6b)s+{a}^{2}-4ab}{(s+a)(s-a)(s-2b)}$

$=\frac{A}{s+a}+\frac{B}{s-a}+\frac{C}{s-2b}+K$

$K=0$

$A=F(s)\ast (s+a)$