- #1

arhzz

- 268

- 52

- Homework Statement
- Solve the ODE

- Relevant Equations
- Laplace Transformation,and Partial Fraction Decomposition

Hello!

Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For example;

$$x'' +25x = cosh(t) $$ with inital values x(0) = 0 and x'(0) = 5.

Now I've transformed it using laplace and pluged in the initial values and this is what I get;

$$ s^2 X(s) - 5 + 25X(s) = \frac{s}{s^2-1} $$

Now this should be correct (so it says in the solution sheet) Now I've tried to isolate X(s) and this is how it went.First I added a 5 to both sides and factored X(s) on the left

$$ X(s) (s^2+25) = \frac{s}{s^2-1} + 5 $$ and now divide with the bracket on the left should give me ;

$$ X(s) = \frac{s}{(s^2-1)(s^2+25} + \frac{5}{s^2+25} $$

Now I think its fairly obvious that the we need partial fraction decomposition here,and that the we have complex zeros. Now here comes the part that I do not undestand.They went ahead and did the following;

$$ X(s) = \frac{s}{(s+1)(s-1)(s^2+25} = \frac{A}{(s+1)} + \frac{B}{(s-1)} + \frac{Cs+D}{s^2+25} $$

I do not understand how they come up with this.I understand where the Cs+D comes from,that is the fact that we have complex roots but where did the 5 go? Why is the second fraction ignored? I would have went with an approach of

$$ X(s) =\frac{s}{(s+1)(s-1)(s^2+25} +\frac{5}{s^2+25} $$ why was the second fraction left out?

Or this example ODE;

$$ x''(t) +4x = e^{2t} $$ with initial values x(0) = 3 and x'(0) = 2 Now after Laplace Transformation and plugging in the initial values I get X(s) to be

$$X(s) =\frac{3s-2}{s^2+4}+ \frac{1}{s-2} \frac{1}{s^2+4} $$

Again we have complex roots and a simple root of 2.Now again I do not understand their approach here;

$$X(s) = \frac{1}{s-2} * \frac{1}{s^2+4} = \frac{A}{s-2} + \frac{Bs+C}{s^2+4} $$

What happened to the 3s-2?I am not understanding how to go around not having a polynom in the nummerator,finding the roots and the right approach(simple roots complex ect..) is not the problem. And another confusing part about this ODE is,when going ahead and calculation the values of A Bs and C we get them to be A = 1/8 B = -1/8 and C = -1/4

Now plugging in those values in the solutions looks like this;

$$ X(s) = 3 -\frac{1s}{8(s^+4)} + 1 - \frac{2}{8(s^2+4)}+\frac{1}{8(s-2)} $$

Now considering their approach I do not see how they get this.The last fraction makes sense they just plugged in the 1/8 but the first 2 are not clicking with me.Also I don't see that the value of C has been used at all.

Thanks for the help and excuse the long post

Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For example;

$$x'' +25x = cosh(t) $$ with inital values x(0) = 0 and x'(0) = 5.

Now I've transformed it using laplace and pluged in the initial values and this is what I get;

$$ s^2 X(s) - 5 + 25X(s) = \frac{s}{s^2-1} $$

Now this should be correct (so it says in the solution sheet) Now I've tried to isolate X(s) and this is how it went.First I added a 5 to both sides and factored X(s) on the left

$$ X(s) (s^2+25) = \frac{s}{s^2-1} + 5 $$ and now divide with the bracket on the left should give me ;

$$ X(s) = \frac{s}{(s^2-1)(s^2+25} + \frac{5}{s^2+25} $$

Now I think its fairly obvious that the we need partial fraction decomposition here,and that the we have complex zeros. Now here comes the part that I do not undestand.They went ahead and did the following;

$$ X(s) = \frac{s}{(s+1)(s-1)(s^2+25} = \frac{A}{(s+1)} + \frac{B}{(s-1)} + \frac{Cs+D}{s^2+25} $$

I do not understand how they come up with this.I understand where the Cs+D comes from,that is the fact that we have complex roots but where did the 5 go? Why is the second fraction ignored? I would have went with an approach of

$$ X(s) =\frac{s}{(s+1)(s-1)(s^2+25} +\frac{5}{s^2+25} $$ why was the second fraction left out?

Or this example ODE;

$$ x''(t) +4x = e^{2t} $$ with initial values x(0) = 3 and x'(0) = 2 Now after Laplace Transformation and plugging in the initial values I get X(s) to be

$$X(s) =\frac{3s-2}{s^2+4}+ \frac{1}{s-2} \frac{1}{s^2+4} $$

Again we have complex roots and a simple root of 2.Now again I do not understand their approach here;

$$X(s) = \frac{1}{s-2} * \frac{1}{s^2+4} = \frac{A}{s-2} + \frac{Bs+C}{s^2+4} $$

What happened to the 3s-2?I am not understanding how to go around not having a polynom in the nummerator,finding the roots and the right approach(simple roots complex ect..) is not the problem. And another confusing part about this ODE is,when going ahead and calculation the values of A Bs and C we get them to be A = 1/8 B = -1/8 and C = -1/4

Now plugging in those values in the solutions looks like this;

$$ X(s) = 3 -\frac{1s}{8(s^+4)} + 1 - \frac{2}{8(s^2+4)}+\frac{1}{8(s-2)} $$

Now considering their approach I do not see how they get this.The last fraction makes sense they just plugged in the 1/8 but the first 2 are not clicking with me.Also I don't see that the value of C has been used at all.

Thanks for the help and excuse the long post

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