Help with ODE using an integrating factor

In summary, the conversation is about solving a Laplace transform problem using integrating factor method. The original answer is correct and the integrating factor is e^2t. The conversation then discusses using an example to explain the process of integrating by parts and arriving at the same solution as the Laplace transform. The conversation ends with the individual being stuck on how to solve the integral.
  • #1
hurcw
23
0
Hi all,

I am doing some Laplace Transforms as part of my HND, i have got an answer for this question
q' +2q = 5sint q(0)=0, t(0)=0

But i need to prove it by means of using an integrating factor method.
My original answer is:-
e^-2t +2sint-cost does this look right?
I also have worked out my Integrating factor to be e^2x but not sure where to go from there.

Can anyone please help me with a few pointers.
 
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  • #2
Do you recognize it's a first order linear diff. eq?
 
  • #3
Your answer is correct. Your integrating factor is correct. Now remember what an integrating factor is. It is a function which, when multiplied by the original equation, makes the equation separable. Review this theory, it is very straightforward. Multiply your equation by the integrating factor. The left side becomes the derivative of q times the integrating factor and thus integrates trivially. The right side becomes the integral of the exponential times the non-homogeneous term sint. This can be done by parts twice (or looking in a table). The result, after applying your condition, is the same as your transform result.
 
  • #4
alan2 thanks for the reply. are you saying my answer to the initial Laplace transform is correct?
Would you be abe to show me what you mean by using another example, i would like to try and work through my question myself and then get the answer possiby verified.
 
  • #5
Any more help would be appreciated guys. Thanks
 
  • #6
Sorry, haven't been here in a couple of days. I assume you don't have an ode book on your shelf. I don't know how to use the advanced equation editor so here goes.
If the equation were separable it would be trivially solvable. So we seek an integrating factor which makes it solvable. Call the factor a. We multiply the equation by a and find a(dq/dt)+2aq=5asint. Now we require that the left side be of the form d(aq)/dt where a=a(t). Then the equation is separable, d(aq)=5asintdt, and we can integrate both sides. But, by the chain rule, d(aq)/dt=adq/dt+qda/dt. If this is to be equal to adq/dt+2aq (from above) then da/dt=2a. This equation for a is itself separable and a=e^2t as you found above. So now we have d(aq)=5asintdt where a is the known function of t which we found. Integrate both sides (the left side is trivially aq) and you find that q=(1/a) times the integral of 5asint. This integral is done by parts twice. Apply the boundary conditions and you have the result that you found above.
 
  • #7
Thanks for the reply, can you explain what you mean by the integrate by parts twice?
I am failing to see how i will end up with the same as i did for my Laplace transform as i don't have a sin & a cos function.
 
  • #9
I have tried to use this example as a guide and just get lost in it all.
am i right in thinking this?:-
∫udv=uv-∫vdu
u=5sin(t)
dv= e(2t)
so:- du=-5cos(t)
v=e(-2t)
Therefore ∫udv=uv-∫vdu
=5e(-2t)sin(t)-∫e(-2t)-5cos(t)
Then i get stuck plus the 2 on the LHS is throwing me.
 

1. What is an integrating factor?

An integrating factor is a function that is used to help solve a differential equation, specifically an ordinary differential equation (ODE). It is multiplied to both sides of the equation to make the equation easier to solve.

2. How do I know when to use an integrating factor?

You should use an integrating factor when you have a first-order linear ODE that cannot be solved using basic methods such as separation of variables or substitution. It is also useful when solving higher-order linear ODEs.

3. How do I find the integrating factor?

The integrating factor can be found by multiplying the equation by an appropriate function. The function is usually determined by the coefficients of the equation. For example, if the equation is y' + P(x)y = Q(x), the integrating factor is e∫P(x)dx.

4. Can you provide an example of solving an ODE using an integrating factor?

Sure. Let's say we have the equation y' + 2xy = x. We can find the integrating factor by multiplying both sides by e∫2xdx, which simplifies to ex2. This gives us the new equation ex2y' + 2xex2y = xex2. We can now solve this using basic integration techniques.

5. Are there any other methods for solving ODEs besides using an integrating factor?

Yes, there are various other methods for solving ODEs, such as separation of variables, substitution, and the method of undetermined coefficients. The method used will depend on the type of ODE and its coefficients. In some cases, it may be necessary to use a combination of methods to solve an ODE.

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