Convert the differential equation

piarepm
2022-01-21
Answered

Convert the differential equation

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Archie Jones

Answered 2022-01-21
Author has **34** answers

Given $u{}^{\u2033}-{u}^{\prime}-2u={e}^{-5t}$

Suppose x=u, y=u',

x'=u' and y'=u''

$\Rightarrow \text{}\text{}{x}^{\prime}=y$ and $y}^{\prime}={u}^{\prime}+2u+{e}^{-5t}+2u+{e}^{-5t}\text{}\text{}\text{}\because \text{}u{}^{\u2033}-{u}^{\prime}-2u={e}^{-5t}\text{}\Rightarrow \text{}u{}^{\u2033}={u}^{\prime$

$\Rightarrow \text{}{x}^{\prime}=y$ and ${y}^{\prime}=y+2x+{e}^{-5t}\text{}\text{}\text{}\because \text{}{u}^{\prime}=y{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}u=x$

Therefore, the system of first order differential equations is,

Answer:

x'=y

$y}^{\prime}=y+2x+{e}^{-5t$

Suppose x=u, y=u',

x'=u' and y'=u''

Therefore, the system of first order differential equations is,

Answer:

x'=y

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The Laplace transform L $\left\{{e}^{-{t}^{2}}\right\}$ exists, but without finding it solve the initial-value problem $y{}^{\u2033}+9y=3{e}^{-{t}^{2}},y\left(0\right)=0,{y}^{\prime}\left(0\right)=0$

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We have the following differential equation

$\cup {}^{\u2033}=1+{\left({u}^{\prime}\right)}^{2}$

i found that the general solution of this equation is

$u=d\text{cosh}((x-\frac{b}{d})$

where b and d are constats

Please how we found this general solution?

i found that the general solution of this equation is

where b and d are constats

Please how we found this general solution?

asked 2022-06-13

My question is to find the solutions to the following

$\frac{df(x)}{dx}={f}^{-1}(x)$

where ${f}^{-1}(x)$ refers to the inverse of the function f. The domain really isn't important, though I am interested in either (-inf, inf) or (0, inf), so if any solutions are known for more restricted domains then they are welcome.

I cannot find any material relating to this type of question in any of my calculus and differential equations textbooks and references; it seems quite unorthodox. Any material which covers this type of diff equation would be wlecome

$\frac{df(x)}{dx}={f}^{-1}(x)$

where ${f}^{-1}(x)$ refers to the inverse of the function f. The domain really isn't important, though I am interested in either (-inf, inf) or (0, inf), so if any solutions are known for more restricted domains then they are welcome.

I cannot find any material relating to this type of question in any of my calculus and differential equations textbooks and references; it seems quite unorthodox. Any material which covers this type of diff equation would be wlecome

asked 2022-01-22

Solve the first-order differential equations:

$({x}^{2}+1)\frac{dy}{dx}=xy$

asked 2022-07-01

Problem 1.7 in G.Teschl ODE and Dynamical Systems asks me to transform the following differential equation into autonomous first-order system:

$\ddot{x}=t\mathrm{sin}(\dot{x})+x$

Transforming the ODE to a system is in this case easy, but whats the usual technique to transform it to an AUTONOMOUS system?

$\ddot{x}=t\mathrm{sin}(\dot{x})+x$

Transforming the ODE to a system is in this case easy, but whats the usual technique to transform it to an AUTONOMOUS system?