Use the Laplace transform to solve the given initial value problem.y"+2y'+5y=0;y(0)=2,y'(0)=−1

OlmekinjP
2021-10-14
Answered

Use the Laplace transform to solve the given initial value problem.y"+2y'+5y=0;y(0)=2,y'(0)=−1

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yagombyeR

Answered 2021-10-15
Author has **92** answers

Result:

asked 2022-04-06

I want to find the laplace inverse of

$s}^{-\frac{3}{2}$

the steps given in the solution manual are as follows:

$\frac{2}{\sqrt{\pi}}\frac{\sqrt{\pi}}{2{s}^{\frac{3}{2}}}=2\sqrt{\frac{t}{\pi}}$

I know the first part$\frac{2}{\sqrt{\pi}}$ is obtained using the gamma function $\mathrm{\Gamma}\left(\frac{3}{2}\right)$ , but not quite sure how the rest is obtained.

the steps given in the solution manual are as follows:

I know the first part

asked 2021-09-22

solve the given initial-value problem.

$y3{y}^{\prime}+2y=\delta (t-1),y\left(0\right)=1,{y}^{\prime}\left(0\right)=0$

asked 2021-09-17

Solve for Y(s), the Laplace transform of the solution y(t) to the initial value problem below.

asked 2022-07-14

A car is travelling at 100 km/h on a level road when it runs out of fuel. Its speed v (in km/h) starts to decrease according to the formula

$\frac{dv}{dt}=-kv\phantom{\rule{1em}{0ex}}(1)$

where k is constant. One kilometre after running out of fuel its speed has fallen to 50 km/h. Use the chain rule substitution

$\frac{dv}{dt}=\frac{dv}{ds}\frac{ds}{dt}=\frac{dv}{ds}v$

to solve the differential equation.

Note: Although I haven't solved it yet, the answers say that this isn't a reasonable model as the velocity is always positive; I didn't make a typo in the question.

What I'm trying to do is solve velocity as a function of displacement (s, in km), velocity as a function of time (t, in hours), and displacement as a function of time (I need these functions for later parts of the question).

So far I've found velocity as a function of displacement (v(s)):

$\frac{dv}{dt}=-k\frac{ds}{dt}\phantom{\rule{1em}{0ex}}\text{(from (1))}$

$\int \frac{dv}{dt}dt=-k\int \frac{ds}{dt}dt$

$v(s)=-ks+C$

$v(0)=100\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}C=100,\text{}v(1)=50\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}k=50$

$v(s)=-50s+100$

Then I've tried to find velocity as a function of time (v(t)), but I've got stuck. I can't find any differential equation I can use to get this, or to get displacement as a function of time (s(t)).

The answer key says $v(t)=100{e}^{-50t}$ and $s(t)=2(1-{e}^{-50t})$

I've solved such questions many times before, but it's been a while so I'm a bit rusty. So, even a hint might be enough for me to realise what to do.

$\frac{dv}{dt}=-kv\phantom{\rule{1em}{0ex}}(1)$

where k is constant. One kilometre after running out of fuel its speed has fallen to 50 km/h. Use the chain rule substitution

$\frac{dv}{dt}=\frac{dv}{ds}\frac{ds}{dt}=\frac{dv}{ds}v$

to solve the differential equation.

Note: Although I haven't solved it yet, the answers say that this isn't a reasonable model as the velocity is always positive; I didn't make a typo in the question.

What I'm trying to do is solve velocity as a function of displacement (s, in km), velocity as a function of time (t, in hours), and displacement as a function of time (I need these functions for later parts of the question).

So far I've found velocity as a function of displacement (v(s)):

$\frac{dv}{dt}=-k\frac{ds}{dt}\phantom{\rule{1em}{0ex}}\text{(from (1))}$

$\int \frac{dv}{dt}dt=-k\int \frac{ds}{dt}dt$

$v(s)=-ks+C$

$v(0)=100\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}C=100,\text{}v(1)=50\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}k=50$

$v(s)=-50s+100$

Then I've tried to find velocity as a function of time (v(t)), but I've got stuck. I can't find any differential equation I can use to get this, or to get displacement as a function of time (s(t)).

The answer key says $v(t)=100{e}^{-50t}$ and $s(t)=2(1-{e}^{-50t})$

I've solved such questions many times before, but it's been a while so I'm a bit rusty. So, even a hint might be enough for me to realise what to do.

asked 2022-01-16

Each of the differential equations is of two different types considered in this chapter-separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutions for each of these equations in two different ways; then reconcile your results.

$\frac{dy}{dx}=3(y+7){x}^{2}$

asked 2020-11-05

Explain First Shift Theorem & its properties?

asked 2021-03-05

Find the laplace transform of the following:

$a){t}^{2}\mathrm{sin}kt$

$b)t\mathrm{sin}kt$