Solving ODEs Using Laplace: Two Challenging Problems

[sandpants]

I have the following 2 problems to solve using Laplace.

1) x'' + 3x' +2x=e^(-t); with x=dx/dt=0 when t = 0
2) x'' - 2x' +10x=e^(2t); with x=0 and dx/dt=1 when t=0
Where x' is dx/dt and x'' is the second derivative against time.

My attempts:

1)Using laplace I get

s²X(s)-x(0)-x'(0)+3sX(s)-x(0)+2X(s)=1/(s+1)

with x(0)=0, x'(0)=0 then

X(s)=1 over (s+1)(s²+3s+2) which is 1/[(s+1)(s+1)(s+2)] or 1/[(s+1)²(s+2)]
Using partial fraction
1/[(s+1)²(s+2)]=A/(s+1)+B/(s+1)²+C/(s+2).

I'll avoid doing the calcs; I may have made a mistake here, but I checked multiple times and didnt find and error.

A=10/12, B=-1/6 and C=-1 so that, after doing inverse laplace

x(t)=5/6e^-t-1/6t*e^-t-e^-2t
But here's the issue. With this, x(0)=/=0. It's -1/6. and dx(0)/dt=1. I can't figure out where I went wrong.

2)The laplace transform is
s² X(s)-x(0)-x'(0)-2sX(s)-x(0)+10X(s)=1/(s-2)
with x'(0)=1 this rearranges to

X(s)(s²-2s+10)=1/(s-2)+1=(s-1)/(s-2)
X(s)=(s-1)/[(s-2)(s²-2s+10)]

This is where I helplessly run around in circles. The quadratic roots are complex, and whilst they discomfort me only slightly, the issue is that I cannot rearrange this into a suitable form where the inverse laplace can be done.

Taken from http://www.therationaltheorist.org/2009/11/fourier-analysis-and-odes.html

 

[Mark44]

I have the following 2 problems to solve using Laplace.

1) x'' + 3x' +2x=e^(-t); with x=dx/dt=0 when t = 0
2) x'' - 2x' +10x=e^(2t); with x=0 and dx/dt=1 when t=0
Where x' is dx/dt and x'' is the second derivative against time.

My attempts:

1)Using laplace I get

s²X(s)-x(0)-x'(0)+3sX(s)-x(0)+2X(s)=1/(s+1)

with x(0)=0, x'(0)=0 then

X(s)=1 over (s+1)(s²+3s+2) which is 1/[(s+1)(s+1)(s+2)] or 1/[(s+1)²(s+2)]
Using partial fraction
1/[(s+1)²(s+2)]=A/(s+1)+B/(s+1)²+C/(s+2).

I'll avoid doing the calcs; I may have made a mistake here, but I checked multiple times and didnt find and error.

A=10/12, B=-1/6 and C=-1 so that, after doing inverse laplace

x(t)=5/6e^-t-1/6t*e^-t-e^-2t
But here's the issue. With this, x(0)=/=0. It's -1/6. and dx(0)/dt=1. I can't figure out where I went wrong.

There's nothing that jumps out at me on how you set things up, but your answer is incorrect.

Using a different method, I got x(t) = (1/3)e^-2t - (1/3)e^-t - (1/3)te^-t.

First place I'd check is the partial fractions decomposition. Can you verify that
$$ \frac{10/12}{s + 1} + \frac{-1/6}{(s + 1)^2} + \frac{-1}{s + 2} = \frac{1}{(s + 1)^2(s + 2)} ?$$

2)The laplace transform is
s² X(s)-x(0)-x'(0)-2sX(s)-x(0)+10X(s)=1/(s-2)
with x'(0)=1 this rearranges to

X(s)(s²-2s+10)=1/(s-2)+1=(s-1)/(s-2)
X(s)=(s-1)/[(s-2)(s²-2s+10)]

This is where I helplessly run around in circles. The quadratic roots are complex, and whilst they discomfort me only slightly, the issue is that I cannot rearrange this into a suitable form where the inverse laplace can be done.

Taken from http://www.therationaltheorist.org/2009/11/fourier-analysis-and-odes.html

 

[sandpants]

I think I've noticed where I've made the mistake: left hand side of the partial fractions.

s+1=A(s+1)^2(s+2)+B(s+1)(s+2)+C(s+1)^3, not 1=A(s+1)^2(s+2)+B(s+1)(s+2)+C(s+1)^3 for which I calculated.

EDIT:

A=-1
B=1
C=1
Is what at I get now.
x(t)=e^(-2t)+te^(-t)-e^(-t) which is equal to 0 on t=0, and the derivative is 0 as well.
However this answer is different from the above.

 

[Mark44]

I think I've noticed where I've made the mistake: left hand side of the partial fractions.

s+1=A(s+1)^2(s+2)+B(s+1)(s+2)+C(s+1)^3, not 1=A(s+1)^2(s+2)+B(s+1)(s+2)+C(s+1)^3 for which I calculated.

EDIT:

A=-1
B=1
C=1
Is what at I get now.
x(t)=e^(-2t)+te^(-t)-e^(-t) which is equal to 0 on t=0, and the derivative is 0 as well.
However this answer is different from the above.

I agree with your new answer. I checked my previous answer, but apparently not well enough, as it was incorrect.

I'll take a look at your other question in a bit.

 

[Mark44]

For the second problem, look at the e^atcos(ωt) and e^atsin(ωt) entries in your table. The expression s² - 2s + 10 can be written as s² - 2s + 1 + 9, and factored as (s - 1)² + 3²

Your solution will involve some linear combination of three functions: e^tsin(3t), e^tcos(3t), and e^2t.

 

[sandpants]

Many thanks.

 

[HallsofIvy]

Can I do my usual rant about how I dislike the whole "Laplace Transform" method? Of course, it is much easier to do both of these 'directly'. The "characteristic" equation for x''+ 3x'+ 2= 0 is r^+ 3r+ 2= (r+ 2)(r+ 1)= 0. which has solutions -2 and -1 so the general solution is x= Ce^{-2t}+ De^{-t}. Then find a particular solution to the entire equation by "undetermined coefficients"- try x= Ate^{-t}.

The second equation has characteristic equation r^2+ 2r-10= (r+ 5)(r- 2)= 0. And you can try x= Ate^{2t} for a specific solution to the equation.

 

[Mark44]

Can I do my usual rant about how I dislike the whole "Laplace Transform" method? Of course, it is much easier to do both of these 'directly'.

I agree, and that's the way I did them.

The "characteristic" equation for x''+ 3x'+ 2= 0 is r^+ 3r+ 2= (r+ 2)(r+ 1)= 0. which has solutions -2 and -1 so the general solution is x= Ce^{-2t}+ De^{-t}. Then find a particular solution to the entire equation by "undetermined coefficients"- try x= Ate^{-t}.

The second equation has characteristic equation r^2+ 2r-10= (r+ 5)(r- 2)= 0.

No, those aren't the factors. As it turns out, the roots of the characteristic equation are complex; namely, r = 1 $\pm 3i$.

And you can try x= Ate^{2t} for a specific solution to the equation.

x_p = Ae^2t will work as a particular solution, as there are no repeated roots of the third-degree characteristic equation (thinking in terms of annihilators).

 

[sandpants]

It was a past examination task.

I'm not particularly impressed by the Laplace transforms. I do see the convenience but I find it is mostly overshadowed by the need to be able to articulate numbers and expressions around very well.