Use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the convolution integral before transforming.(Write your answer as a function of s.)

$L\{{e}^{-t}\cdot {e}^{t}\mathrm{cos}\left(t\right)\}$

CMIIh
2021-01-05
Answered

Use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the convolution integral before transforming.(Write your answer as a function of s.)

$L\{{e}^{-t}\cdot {e}^{t}\mathrm{cos}\left(t\right)\}$

You can still ask an expert for help

opsadnojD

Answered 2021-01-06
Author has **95** answers

Step 1

Consider the provided question,

We have to find$L\{{e}^{-t}\cdot {e}^{t}\mathrm{cos}\left(t\right)\}$
Step 2

Now, the given Laplace transform is find as,

$L\{{e}^{-t}\cdot {e}^{t}\mathrm{cos}\left(t\right)\}=L\left\{{e}^{-t}\right\}+L\{{e}^{t}\mathrm{cos}\left(t\right)\}$

$=\left(\frac{1}{s+1}\right)\cdot \frac{s-1}{{(s-1)}^{2}+{1}^{2}}[Use,L\left({e}^{at}\right)=\frac{1}{s-a}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}L({e}^{at}\mathrm{cos}bt)=\frac{s-a}{{(s-a)}^{2}+{b}^{2}}]$

$=\frac{s-1}{(s+1)[{(s-1)}^{2}+1]}$

Thus ,$L\{{e}^{-t}\cdot {e}^{t}\mathrm{cos}\left(t\right)\}=\frac{s-1}{(s+1)[{(s-1)}^{2}+1]}$

Consider the provided question,

We have to find

Now, the given Laplace transform is find as,

Thus ,

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I am trying to take the laplace transform of $\mathrm{cos}\left(t\right)u(t-\pi )$ . Is it valid for me to treat it as $((\mathrm{cos}\left(t\right)+\pi )-\pi )u(t-\pi )$ and treat $\mathrm{cos}\left(t\right)-\pi$ as f(t) and use the 2nd shifting property, or is this not the correct procedure?

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Solve the linear equations by considering y as a function of x, that is

$y=y(x)$

$y}^{\prime}-2xy={e}^{{x}^{2}$

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Find the inverse Laplace transform of $F\left(s\right)=\frac{7s-6}{{s}^{2}-4}$

asked 2022-01-18

Differential equation $f{}^{\u2033}\left(x\right)+\frac{(n-1){\left({f}^{\prime}\left(x\right)\right)}^{2}}{\mathrm{sin}h\left(x\right)}=0$

asked 2022-05-23

In my differential equations book, I have found the following:

Let ${P}_{0}(\frac{dy}{dx}{)}^{n}+{P}_{1}(\frac{dy}{dx}{)}^{n-1}+{P}_{2}(\frac{dy}{dx}{)}^{n-2}+......+{P}_{n-1}(\frac{dy}{dx})+{P}_{n}=0$ be the differential equation of first degree 1 and order n (where ${P}_{i}$ $\mathrm{\forall}$ i $\in 0,1,2,...n$ are functions of x and y).

Assuming that it is solvable for p, it can be represented as:

$[p-{f}_{1}(x,y)][p-{f}_{2}(x,y)][p-{f}_{3}(x,y)]........[p-{f}_{n}(x,y)]=0$

equating each factor to Zero, we get n differential equations of first order and first degree.

$[p-{f}_{1}(x,y)]=0,\text{}[p-{f}_{2}(x,y)]=0,\text{}[p-{f}_{3}(x,y)]=0,\text{}........[p-{f}_{n}(x,y)]=0$

Let the solution to these n factors be:

${F}_{1}(x,y,{c}_{1})=0,\text{}{F}_{2}(x,y,{c}_{2})=0,\text{}{F}_{3}(x,y,{c}_{3})=0,\text{}........{F}_{n}(x,y,{c}_{n})=0$

Where ${c}_{1},{c}_{2},{c}_{3}.....{c}_{n}$ are arbitrary constants of integration. Since all the c’s can have any one of an infinite number of values, the above solutions will remain general if we replace ${c}_{1},{c}_{2},{c}_{3}.....{c}_{n}$ by a single arbitrary constant c. Then the n solutions (4) can be re-written as

${F}_{1}(x,y,c)=0,\text{}{F}_{2}(x,y,c)=0,\text{}{F}_{3}(x,y,c)=0,\text{}........{F}_{n}(x,y,c)=0$

They can be combined to form the general solution as follows:

${F}_{1}(x,y,c)\text{}{F}_{2}(x,y,c)\text{}{F}_{3}(x,y,c)\text{}........{F}_{n}(x,y,c)=0\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}(1)$

Now, my question is, whether equation (1) is the most general form of solution to the differential equation.I think the following is the most general form of solution to the differential equation :

${F}_{1}(x,y,{c}_{1})\text{}{F}_{2}(x,y,{c}_{2})\text{}{F}_{3}(x,y,{c}_{3})\text{}........{F}_{n}(x,y,{c}_{n})=0\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}(2)$

If (1) is the general solution, the constant of integration can be found out by only one IVP say, $y(0)=0$. So, one IVP will give the particular solution. If (2) is the general solution, one IVP might not be able to give the particular solution to the problem.

Let ${P}_{0}(\frac{dy}{dx}{)}^{n}+{P}_{1}(\frac{dy}{dx}{)}^{n-1}+{P}_{2}(\frac{dy}{dx}{)}^{n-2}+......+{P}_{n-1}(\frac{dy}{dx})+{P}_{n}=0$ be the differential equation of first degree 1 and order n (where ${P}_{i}$ $\mathrm{\forall}$ i $\in 0,1,2,...n$ are functions of x and y).

Assuming that it is solvable for p, it can be represented as:

$[p-{f}_{1}(x,y)][p-{f}_{2}(x,y)][p-{f}_{3}(x,y)]........[p-{f}_{n}(x,y)]=0$

equating each factor to Zero, we get n differential equations of first order and first degree.

$[p-{f}_{1}(x,y)]=0,\text{}[p-{f}_{2}(x,y)]=0,\text{}[p-{f}_{3}(x,y)]=0,\text{}........[p-{f}_{n}(x,y)]=0$

Let the solution to these n factors be:

${F}_{1}(x,y,{c}_{1})=0,\text{}{F}_{2}(x,y,{c}_{2})=0,\text{}{F}_{3}(x,y,{c}_{3})=0,\text{}........{F}_{n}(x,y,{c}_{n})=0$

Where ${c}_{1},{c}_{2},{c}_{3}.....{c}_{n}$ are arbitrary constants of integration. Since all the c’s can have any one of an infinite number of values, the above solutions will remain general if we replace ${c}_{1},{c}_{2},{c}_{3}.....{c}_{n}$ by a single arbitrary constant c. Then the n solutions (4) can be re-written as

${F}_{1}(x,y,c)=0,\text{}{F}_{2}(x,y,c)=0,\text{}{F}_{3}(x,y,c)=0,\text{}........{F}_{n}(x,y,c)=0$

They can be combined to form the general solution as follows:

${F}_{1}(x,y,c)\text{}{F}_{2}(x,y,c)\text{}{F}_{3}(x,y,c)\text{}........{F}_{n}(x,y,c)=0\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}(1)$

Now, my question is, whether equation (1) is the most general form of solution to the differential equation.I think the following is the most general form of solution to the differential equation :

${F}_{1}(x,y,{c}_{1})\text{}{F}_{2}(x,y,{c}_{2})\text{}{F}_{3}(x,y,{c}_{3})\text{}........{F}_{n}(x,y,{c}_{n})=0\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}(2)$

If (1) is the general solution, the constant of integration can be found out by only one IVP say, $y(0)=0$. So, one IVP will give the particular solution. If (2) is the general solution, one IVP might not be able to give the particular solution to the problem.

asked 2021-12-27

A thermometer reading ${75}^{\circ}F$ is taken out where the temperature is ${20}^{\circ}F$ . The reading is ${30}^{\circ}F$ 4 min later.

A. What is the value of k in four decimal places?

B. Find the thermometer reading 7 min after the thermometer was brought outside.

C. Find the time taken for the reading to drop from${75}^{\circ}F$ to within a half degree of the air temperature.

A. What is the value of k in four decimal places?

B. Find the thermometer reading 7 min after the thermometer was brought outside.

C. Find the time taken for the reading to drop from

asked 2021-01-15

Show that the first order differential equation $(x+1){y}^{\prime}-3y=(x+1{)}^{5}$ is of the linear type.

Hence solve for y given that y = 1.5 when x = 0

Hence solve for y given that y = 1.5 when x = 0