Solve ODE with direct integration

In summary, the conversation is about the difficulties the speaker is having with integrating functions and the confusion surrounding the use of "variable - dummy variable" in the process. They mention trying to plug it into the original function and not knowing how to take the derivative of an integral with a mix of dummy and real variables. They also mention the Cauchy formula for repeated integration as a possible solution.
  • #1
veneficus5
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Homework Statement


Latex takes me forever so I'm going to take a picture



Homework Equations





The Attempt at a Solution



I'm having issues with integrating functions. There seems to be this (x-x0) term that crops up everywhere. Last time it was (t - tau). It's always (variable - dummy variable) so far, but I don't understand why. I also tried plugging it into the original, but I didn't know how to take the derivative of an integral with a mix of dummy and real variables. I guess this is something I should have learned in calculus but never did?

Thanks a lot!
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  • #2
Recall from elementary calculus the Cauchy formula for repeated integration, which follows from integrationby parts and allows one to write a repeated integral as a single integral. In other words you have found two equivelant ways to express the answer.
 

1. How do you solve an ODE using direct integration?

To solve an ODE using direct integration, you need to first separate the variables in the ODE and then integrate both sides. This will give you the general solution of the ODE, which can be further simplified by applying initial conditions.

2. What is the advantage of using direct integration to solve ODEs?

The advantage of using direct integration is that it is a straightforward and systematic method for solving ODEs. It involves simple mathematical operations and does not require any advanced techniques or methods.

3. Can direct integration be used to solve all types of ODEs?

No, direct integration can only be used to solve separable ODEs, which are those that can be written in the form of dy/dx = f(x)g(y). For other types of ODEs, different methods such as substitution or variation of parameters may be required.

4. How do you determine the particular solution when solving an ODE with direct integration?

To determine the particular solution, you need to apply the initial conditions to the general solution obtained through direct integration. This will give you a specific solution that satisfies both the ODE and the initial conditions.

5. Are there any limitations to using direct integration to solve ODEs?

Yes, direct integration can only be used for ODEs with separable variables. It cannot be used for non-separable ODEs, as well as higher-order ODEs. In addition, it may not always give the most efficient or accurate solution compared to other methods.

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