Determine a unique solution of the separable differential equa- tion that satisfies the given initial condition.

$\frac{{d}^{2}y}{{dt}^{2}}=\frac{dy}{dt}\frac{dy}{dt}\text{}\left(0\right)=2,\text{}y\left(0\right)=3$

ezpimpin6988ok1n
2022-03-24
Answered

Determine a unique solution of the separable differential equa- tion that satisfies the given initial condition.

$\frac{{d}^{2}y}{{dt}^{2}}=\frac{dy}{dt}\frac{dy}{dt}\text{}\left(0\right)=2,\text{}y\left(0\right)=3$

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pypberissootcu

Answered 2022-03-25
Author has **14** answers

Given Differential equation is

$\frac{{d}^{2}y}{{dt}^{2}}=\frac{dy}{dt}$

Boundary Condition is

$\frac{dy}{dt}\left(0\right)=2$ and $y\left(0\right)=3$

Solution:

$\frac{{d}^{2}y}{{dt}^{2}}=\frac{dy}{dt}$

Substitute$\frac{dy}{dt}={y}^{\prime}$

So,

$y{}^{\u2033}={y}^{\prime}$

$y{}^{\u2033}-{y}^{\prime}=0$

This is second order linear homogeneous ordinary differential equation

Auxillary equation is

${m}^{2}-m=0$

$m(m-1)=0$

Roots are$m=0$ and $m=1$

Roots are real and district

So solution is

$y\left(t\right)={C}_{1}{e}^{{m}_{1}t}+{C}_{2}{e}^{{m}_{2}t}$

Here put${m}_{1}=0$ and ${m}_{2}=1$

So solution is

$y={C}_{1}{e}^{0\cdot t}+{C}_{2}{e}^{1\cdot t}$

$y\left(t\right)={C}_{1}+{C}_{2}{e}^{t}$

Use Given Initial Condition

$y\left(0\right)={C}_{1}+{C}_{2}{e}^{0}={C}_{1}+{C}_{2}=3$

$y\left(t\right)={C}_{1}+{C}_{2}{e}^{T}$

$y}^{\prime}\left(t\right)={C}_{2}{e}^{t$

$y0)={C}_{2}=2$

$\Rightarrow {C}_{1}+2=3$

${C}_{1}=3-2=1$

$\Rightarrow {C}_{1}=1$

Put${C}_{1}=1$ and $C}_{2$ in y(t)

$y\left(t\right)=1+2{e}^{t}$

$y\left(t\right)=1+2{e}^{t}$ is solution of given Differential Equation

Boundary Condition is

Solution:

Substitute

So,

This is second order linear homogeneous ordinary differential equation

Auxillary equation is

Roots are

Roots are real and district

So solution is

Here put

So solution is

Use Given Initial Condition

Put

Jeffrey Jordon

Answered 2022-03-31
Author has **2262** answers

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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 2022-05-19

I have a question on the particular case of the Cauchy theorem for a first order differential equation with separable variables.

${y}^{\prime}(x)=a(x)b(y(x))$

If I impose:

$a(x)$ continuous in a interval $I$

$b(y(x))$ continuous and with continuous derivative in a interval $J$

Can I say that there is only one solution in all the interval I that satisfies the condition of passage through a point $({x}_{0},{y}_{0})\in I\times J$?

Or, instead of the continuity of the derivative of $b(y(x))$, should I impose that the derivative of $b(y(x))$ is limited on $J$?

Can anyone suggest me the correct conditions to impose in this particular case?

Thanks in adivce

${y}^{\prime}(x)=a(x)b(y(x))$

If I impose:

$a(x)$ continuous in a interval $I$

$b(y(x))$ continuous and with continuous derivative in a interval $J$

Can I say that there is only one solution in all the interval I that satisfies the condition of passage through a point $({x}_{0},{y}_{0})\in I\times J$?

Or, instead of the continuity of the derivative of $b(y(x))$, should I impose that the derivative of $b(y(x))$ is limited on $J$?

Can anyone suggest me the correct conditions to impose in this particular case?

Thanks in adivce

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Consider a bacterial population that grows according to the function $f\left(t\right)=500{e}^{0.05t}$ measured in minutes. After 4 hours, how many germs are present in the population? When will the population of bacteria reach 100 million?

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Find the Laplace transform of $f\left(t\right)=t{e}^{-t}\mathrm{sin}\left(2t\right)$

Then you obtain$F\left(s\right)=\frac{4s+a}{{({(s+1)}^{2}+4)}^{2}}$

Please type in a = ?

Then you obtain

Please type in a = ?

asked 2022-07-16

How can I solve this first order linear differential equation?

${y}^{\prime}=1-\frac{2}{x+y}$

I have tried turning it into an inexact differential equation, but I get an integration factor $\mu (x,y)$ and I don't know how to apply it.

${y}^{\prime}=1-\frac{2}{x+y}$

I have tried turning it into an inexact differential equation, but I get an integration factor $\mu (x,y)$ and I don't know how to apply it.

asked 2022-07-04

Express the differential equation

${y}^{\u2034}-6{y}^{\u2033}-{y}^{\prime}+6y=0$

as a system of first order equations i.e. a matrix equation of the form

$A(\overrightarrow{x}{)}^{\prime}=0$

where

$\overrightarrow{x}\text{is the vector}\left[\begin{array}{r}{x}_{1}\\ \text{}{x}_{2}\\ \text{}{x}_{3}\end{array}\right].$

${y}^{\u2034}-6{y}^{\u2033}-{y}^{\prime}+6y=0$

as a system of first order equations i.e. a matrix equation of the form

$A(\overrightarrow{x}{)}^{\prime}=0$

where

$\overrightarrow{x}\text{is the vector}\left[\begin{array}{r}{x}_{1}\\ \text{}{x}_{2}\\ \text{}{x}_{3}\end{array}\right].$