Help me solve this differential equation

In summary: Well... considering it is the right answer... it should match the question. What is the method for getting that answer.It can be instructive to check that the "correct answer" matches the question ... did you try? Did you try the transform on your answer?Going forward:Did you try completing the square in s^2 in the denominator.hint: complete the square in s^2 in the denominator.I don't know what that means.
  • #1
grandpa2390
474
14

Homework Statement


find the inverse laplace transform.

1/(s^3 + 7s)

Homework Equations


sin(kt) = k/(s^2 + k^2)
cos(kt) = s/(s^2 + k^2)

The Attempt at a Solution



so this one is different from all the others I have done because it involves an imaginary number and I am not sure what the rule is that changes the problem.
somehow the answer becomes 1/7 - cos(sqrt(7)*t ) / 7

but I am getting positive sine instead of negative cosine and 7i in the denominator instead of 7. does the i reverse sine to cosine or something?
 
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  • #2
somehow the answer becomes 1/7 - cos(sqrt(7)*t ) / 7
... try taking the Laplace transform of the answer and see if it matches the question.
 
  • #3
Simon Bridge said:
... try taking the Laplace transform of the answer and see if it matches the question.

Well... considering it is the right answer... it should match the question. What is the method for getting that answer.
 
  • #4
It can be instructive to check that the "correct answer" matches the question ... did you try? Did you try the transform on your answer?

Going forward:
Did you try completing the square in s^2 in the denominator.
 
  • #5
Simon Bridge said:
hint: complete the square in s^2 in the denominator.
I don't know what that means.

I set up my partial fraction. A(s^2+7) + B(s) = 1
s=0 and sqrt(-7)
A= 1/7 and B= 1/sqrt(-7)

1/(7s) + 1/(sqrt(-7)*(s^2+7))

1/7 * 1/s = 1/7
1/sqrt(-7)sqrt(7) * sqrt(7)/(s^2 + 7) = 1/(7i) * sin(sqrt(7)t)

1/7 + sin(sqrt(7)t)/7i
but that is wrong I should get 1/7 - cos(sqrt(7)t)/7
 
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  • #6
Try getting the denominator in form: ##(s^2 + a^2)^2##

Put it another way: please show your working.
 
  • #7
Simon Bridge said:
Try getting the denominator in form: ##(s^2 + a^2)^2##

Put it another way: please show your working.

there I posted my work.
 
  • #8
Simon Bridge said:
Try getting the denominator in form: ##(s^2 + a^2)^2##

so you are saying it is ok to multiply the initial problem by s/s and then I won't have to worry about i.

edit: no that didn't work

I do not have a clue.
 
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  • #9
I set up my partial fraction. A(s^2+7) + B(s) = 1
s=0 and sqrt(-7)
A= 1/7 and B= 1/sqrt(-7)

1/(7s) + 1/(sqrt(-7)*(s^2+7))

1/7 * 1/s = 1/7
1/sqrt(-7)sqrt(7) * sqrt(7)/(s^2 + 7) = 1/(7i) * sin(sqrt(7)t)

1/7 + sin(sqrt(7)t)/7i
but that is wrong I should get 1/7 - cos(sqrt(7)t)/7
... In the partial fractions decomposition: don't you need a Bs+C in the numerator of the quadratic denominator?
I found it helped to put a^2=7 and just write everything out in terms of a.
 
  • #10
I'll try yahoo answers.
Simon Bridge said:
... In the partial fractions decomposition: don't you need a Bs+C in the numerator of the quadratic denominator?
I found it helped to put a^2=7 and just write everything out in terms of a.

that might be it then. I should have done Bs+C. and that would put the s in the numerator required for the cosine function. but will it get rid of the i?
 
  • #11
If I read you correctly: the "i" comes from the imaginary root of the quadratic - nothing to do with the decomposition.
In general, the polynomial in the numerator should be, at most, one order less than the polynomial in the denominator.
 
  • #12
grandpa2390 said:

Homework Statement


find the inverse laplace transform.

1/(s^3 + 7s)

Homework Equations


sin(kt) = k/(s^2 + k^2)
cos(kt) = s/(s^2 + k^2)

The Attempt at a Solution



so this one is different from all the others I have done because it involves an imaginary number and I am not sure what the rule is that changes the problem.
somehow the answer becomes 1/7 - cos(sqrt(7)*t ) / 7

but I am getting positive sine instead of negative cosine and 7i in the denominator instead of 7. does the i reverse sine to cosine or something?
Use either (1) If ##f(t) \leftrightarrow g(s)## then ##\int_0^t f(\tau) \, d\tau \leftrightarrow \frac{g(s)}{s}##; or (2) a partial-fraction expansion of ##\frac{1}{s^3 + 7s}##.
 
  • #13
... will it get rid of the i?
I had a think further - you tried a "partial fractions" of form:
$$\frac{A}{s}+\frac{B}{s^2+7}$$... and discovered the relations for A and B were: $$As^2 + 7A +Bs = 1\\ \implies As^2=0,\; 7A=1,\; Bs=0$$ ... the first tells you that A=0 while the second tells you that A=1/7 ... which is an inconsistency, which should have been a hint that you'd done something wrong.
By working out the roots of s^2+7 you were trying to find the values of s that would make the relations true(?) ... which is an OK strategy except that the decomposition is supposed to work for all values of s not just a few of them, so the strategy failed. How you incorporated the roots into the partial fractions is still beyond me - but I suspect it is enough to notice that you had not understood what the partial fractions method is supposed to do.
 
Last edited:
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1. What is a differential equation?

A differential equation is an equation that relates a function to its derivatives. It describes the rate of change of a system or process over time.

2. How do I solve a differential equation?

There are various methods for solving differential equations, including separation of variables, substitution, and using specific formulas for common types of differential equations. It is important to first identify the type of differential equation and then choose the appropriate method for solving it.

3. Why is solving a differential equation important?

Differential equations are used in many fields of science, such as physics, engineering, and economics, to model and understand real-world systems and processes. Solving these equations allows us to make predictions and gain insights into the behavior of these systems.

4. Can I solve a differential equation without knowing calculus?

No, a basic understanding of calculus is necessary to solve differential equations. These equations involve derivatives, which are fundamental concepts in calculus.

5. Are there software programs that can help me solve differential equations?

Yes, there are many software programs, such as Mathematica, MATLAB, and Maple, that can solve differential equations numerically or symbolically. However, it is still important to have a basic understanding of the underlying concepts and methods for solving differential equations.

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