Expressing the Solution to a DE in the Form ce\alphax*sin(\betax+\gamma)

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In summary, the DE y''-2y'+2y=0 can be solved by plugging in the values of y and solving for lambda, which results in lambda = 1+ i or 1-i. A particular solution to the DE is y(x)= e(1+i)x, and when expanded using Euler's formula, the general solution is y(x) = k1excos(x)+k2exsin(x). To express the answer in the form ce\alphax*sin(\betax+\gamma), alpha=1 and beta=1 while gamma can be defined as cos(gamma)=k2 and sin(gamma)=k1. However, gamma is an arbitrary constant, just like c.
  • #1
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Homework Statement


Solve the DE:
y''-2y'+2y=0

and express your answer in the form

ce[tex]\alpha[/tex]x*sin([tex]\beta[/tex]x+[tex]\gamma[/tex])

where alpha = ;
and beta =

Homework Equations



y = e[tex]\lambda[/tex]x

The Attempt at a Solution



When you plug in the values of y into the DE and solve for lambda you get
[tex]\lambda[/tex] = 1+ i or 1-i

A particular solution to the DE is
y(x)= e(1+i)x

when this is expanded using Euler's formula to solve the general solution to the DE, I get

y(x) = k1excos(x)+k2exsin(x)

When looking at the form of the solution they want me to express my answer in I feel like that are asking for

k2exsin(x) = ce[tex]\alpha[/tex]x*sin([tex]\beta[/tex]x+[tex]\gamma[/tex])

alpha = 1 but how do I get beta? I don't think I can express my answer with gamma

Thanks!
 
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  • #2
If you rewrite

[tex] y(x) = \sqrt{k_1^2 + k_2^2}~ e^x \left( \frac{k_1}{\sqrt{k_1^2 + k_2^2}} \cos(x)+\frac{k_2}{\sqrt{k_1^2 + k_2^2}} \sin(x) \right), [/tex]

can you guess at how the angle [tex]\gamma[/tex] is defined?
 
  • #3
To me it seems that beta=1 and gamma is such that cos(gamma)=k2, sin(gamma)=k1. All these having the trigonemetric identity sin(bx+g)=sin(g)cos(bx)+cos(g)sin(bx).
 
  • #4
isn't γ an entirely arbitrary constant, exactly like c ? :confused:
 

1. What is the meaning of "express DE in this form"?

"DE" stands for "differential equation". Expressing a differential equation in a certain form means rewriting it in a specific mathematical format, such as a polynomial or exponential equation. This can make it easier to solve or analyze the equation.

2. Why is it important to express DE in a specific form?

Different forms of differential equations can reveal different insights and properties of the equation. Some forms may be easier to solve or understand, while others may provide important information about the behavior of the system being modeled.

3. How do I express a DE in a specific form?

This process may vary depending on the specific form you are trying to achieve. Generally, it involves manipulating the equation using mathematical operations and transformations until it matches the desired form. It may also involve using known techniques and formulas specific to the desired form.

4. Can any DE be expressed in a specific form?

No, not every differential equation can be expressed in a specific form. Some equations may not have a closed-form solution or may require advanced mathematical techniques to express in a specific form. It is important to understand the limitations and assumptions of the desired form when attempting to express a DE in that form.

5. How can expressing DE in a specific form be useful?

Expressing a differential equation in a specific form can help in solving the equation, understanding its behavior, and making predictions about the system being modeled. It can also provide a more concise and elegant representation of the equation, making it easier to communicate and work with.

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