Finding the solution to an IVP Problem. Basic Differential Equations problem.

In summary, the conversation is about solving an initial value problem involving ty'+7y=2t^2 e^2t with the initial condition y(1)=7. The question is whether the equation is linear and in what interval the solutions exist. The person mentions being stuck on the problem due to a complicated integration and provides a link to the integration answer. The conversation then turns to discussing different methods of solving the problem, including using integration by parts. Eventually, the person is able to solve the problem correctly.
  • #1
cyturk
8
0
I am trying to solve an IVP problem and I seem to be stuck on it because I am getting an integration that seems very complicated and I think I messed up on it, I have my work so far below.

Homework Statement



Find the solution to the IVP

ty^'+7y=2t^2 e^2t, y(1)=7

Is this equation linear? Determine in what interval the solutions exist.


Homework Equations





The Attempt at a Solution



rs1nk.jpg


The image has my work so far, as you can see the integration for (t^7)(2te^2t) is a beast and that is why I think I am wrong so far. Here is the integration answer http://www.wolframalpha.com/input/?i=integrate+%28t^7%29%282te^%282t%29%29.
 
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  • #2
The primitives of (t^8)exp(2t) are on the form P(x)exp(2t) were P(x) is a 8th degree polynomial.
Derive this function and find the coefficients of the polynomial by indentification with (t^8)exp(2t)
 
  • #3
You started out very well, and from:
[tex]
\frac{d}{dt}(t^{7}y)=2t^{8}e^{2t}
[/tex]
I think you have made an error, you can do two things: 1) You can do an indefinite integration and add an integration constant and find that constand by using the initial condition or 2) integrate from 1 to t both sides and use Y(1)=7.
 
  • #4
Integrate 2t8e2t using integration by parts- 8 times!
 
  • #5
HallsofIvy said:
Integrate 2t8e2t using integration by parts- 8 times!

Thanks for everyone else and this is what I ended up doing. It was the correct way of solving the problem even though it was a little bit of a hassle! :)
 

1. What is an IVP problem in differential equations?

An IVP (initial value problem) in differential equations is a mathematical problem that involves finding a solution to an ordinary differential equation (ODE) with a given initial condition. The initial condition specifies the value of the dependent variable at a specific point in the independent variable's domain.

2. How do I solve an IVP problem in differential equations?

To solve an IVP problem in differential equations, you can use analytical or numerical methods. Analytical methods involve using known techniques such as separation of variables or variation of parameters to find a closed-form solution. Numerical methods involve approximating the solution through a series of calculations using a computer.

3. What are some common techniques for solving IVP problems?

Some common techniques for solving IVP problems include Euler's method, Runge-Kutta methods, and the shooting method. These techniques involve approximating the solution using a step-by-step process and adjusting the initial conditions until a desired accuracy is achieved.

4. How do I know if my solution to an IVP problem is correct?

A solution to an IVP problem in differential equations is considered correct if it satisfies both the differential equation and the initial condition. This can be verified by substituting the solution into the equation and checking that it holds true. Additionally, numerical solutions can be compared to analytical solutions for accuracy.

5. Can IVP problems have multiple solutions?

Yes, IVP problems can have multiple solutions. This is because differential equations are often nonlinear and can have infinitely many solutions. However, a unique solution can be found by specifying additional conditions, such as a boundary condition, to narrow down the possible solutions.

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