Solving an Initial Value Problem: 9r^2-12r+4=0

In summary, the given initial value problem is solved by finding the general solution, which is given by y(t)=c_1e^2/3t +c_2te^2/3t. This solution is obtained by factoring the characteristic equation and using the double root to find the second solution. The initial conditions are then used to find the values of the constants c_1 and c_2. Using the initial conditions, the specific solution to the problem is found to be y(t)=2e^2/3t -e^2/3t.
  • #1
Punchlinegirl
224
0
Solve the given initial value problem.
9y"-12y'+4y=0 y(0)=2 y'(0)=-1

9r^2-12r +4=0
(3r-2)^2
so r=2/3
so my general solution would be y(t)=c_1e^2/3t +c_2e^2/3t
Whenever I try to use the initial conditions I can't get them to work. One disappears.
Is my general solution even right?
 
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  • #2
You only have one linearly independent solution there. When you get a double root like that, the other solution is given by, in this case, [itex]t e^{2/3 t} [/tex]. You can verify this by plugging it into the DE.
 
  • #3
I get it.. thanks a lot!
 

FAQ: Solving an Initial Value Problem: 9r^2-12r+4=0

1. What is an Initial Value Problem (IVP)?

An Initial Value Problem is a mathematical problem that involves finding the solution to a differential equation given a set of initial conditions, usually in the form of values for the dependent variable and its derivatives at a specific point.

2. How do you solve an Initial Value Problem?

To solve an Initial Value Problem, you need to first identify the differential equation and the given initial conditions. Then, you can use different methods such as separation of variables, substitution, or Laplace transforms to find the solution. Finally, you can verify the solution by checking if it satisfies the initial conditions.

3. What is the purpose of solving an Initial Value Problem?

Solving an Initial Value Problem allows us to find the exact solution to a differential equation for a given set of initial conditions. This is useful in many applications, such as predicting the behavior of a physical system or modeling real-world phenomena.

4. How do you know if a solution to an Initial Value Problem is unique?

The solution to an Initial Value Problem is unique if it satisfies two conditions: 1) the differential equation is well-defined and continuous for all values of the independent variable, and 2) the initial conditions are specific and unique. If both conditions are met, the solution will be unique.

5. Can an Initial Value Problem have multiple solutions?

No, an Initial Value Problem can only have one unique solution if the two conditions mentioned in the previous answer are satisfied. If there are multiple solutions, it means that one or both of the conditions have not been met, or there is an error in the solution.

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