Exact Solution for Linear ODE with Initial Condition

  • Thread starter longball
  • Start date
  • Tags
    Ode
In summary, "ODE using exact methods" refers to the mathematical approach of finding an analytical solution to ordinary differential equations (ODEs) using known mathematical functions. This is different from numerical methods which use algorithms to approximate the solution. Some examples of ODEs that can be solved using exact methods include the logistic equation, simple harmonic oscillator equation, and Bernoulli equation. The steps involved in using exact methods include identifying the type of ODE, manipulating it into a solvable form, and integrating to find the solution. While exact methods provide a precise understanding of the system, they have limited applicability and can be time-consuming and require advanced mathematical skills.
  • #1
longball
5
0
Consider the ODE

dy/dx + 0.6y = 0.5e^(-(1.1)x) , y(0) = 4

solve the ODE subject to the given condition using exact methods and evaluate the solution y for x = 0.0 (0.05) 0.5, (i.e from x = 0 to x = 0.5 in steps of 0.05)

I am terrible with ODEs and would greatly appreciate help in rearranging and staring off this question. I am sure its not terribly difficult but to me it is. Thanks in advance.
 
Physics news on Phys.org
  • #2
Well, it's linear so start by finding an integrating factor.
 

1. What is the concept of "ODE using exact methods"?

"ODE using exact methods" refers to the mathematical approach used to solve ordinary differential equations (ODEs) by finding an exact analytical solution. This means that the solution is expressed in terms of known mathematical functions, rather than approximated using numerical methods.

2. How do exact methods differ from numerical methods in solving ODEs?

Exact methods involve finding an analytical solution to an ODE, while numerical methods use algorithms to approximate the solution. Exact methods are generally more precise and provide a deeper understanding of the behavior of the system described by the ODE, but they can only be applied to a limited number of ODEs with specific forms.

3. What are some examples of ODEs that can be solved using exact methods?

Some examples of ODEs that can be solved using exact methods include the logistic equation, the simple harmonic oscillator equation, and the Bernoulli equation. These equations have specific forms that allow for an analytical solution to be found using exact methods.

4. What are the steps involved in using exact methods to solve an ODE?

The first step is to identify the type of ODE and determine if it can be solved using exact methods. Then, the ODE is manipulated into a form that allows for an analytical solution to be found. This may involve separating variables, using substitution, or applying an integrating factor. Finally, the solution is found by integrating both sides of the equation and solving for the dependent variable.

5. What are the advantages and limitations of using exact methods to solve ODEs?

The main advantage of using exact methods is that they provide a precise and analytical solution to the ODE, allowing for a deeper understanding of the system's behavior. However, they can only be applied to a limited number of ODEs and may not always be feasible for complex systems. Additionally, the process of finding an exact solution can be time-consuming and may require advanced mathematical skills.

Similar threads

Solving a first order differential equation with initial conditions
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Differential Equation ODE Solution help.
    • Calculus and Beyond Homework Help
Replies
3
Views
845
Two Solutions for y' = 5x^3(y-1)^1/5 with Initial Condition y(0)=1
    • Calculus and Beyond Homework Help
Replies
11
Views
953
Show that ODE is homogeneous, but I don't think it is
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
POTW Solve Linear System of ODEs: x',y' w/ Initial Conditions x(0),y(0)
    • Math POTW for University Students
Replies
10
Views
1K
Understanding First Order ODEs and Intersection of Curves
    • Calculus and Beyond Homework Help
Replies
4
Views
672
Solve an ODE using exact methods?
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Perturbation Methods: Asymptotic Matching Question
    • Calculus and Beyond Homework Help
Replies
2
Views
704
4th order RK to solve 2nd order ODE
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Proving basic linear ODE results
    • Calculus and Beyond Homework Help
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top