Linear ODE Solutions Without Initial Conditions and the Arbitrary Constant C

In summary, if a first order ODE has a solution u and another solution w is obtained without initial conditions, then w can be considered as a function of an arbitrary constant C. It is possible for u to equal w(C) for a specific value of C, which can be found using the boundary conditions. However, in order to have a unique solution to the ODE, some assumptions about its form must be made. Without these assumptions, the ODE may have multiple solutions.
  • #1
jdstokes
523
1
Suppose I already have a solution [itex]u[/itex] to a first order ODE.

If I try to solve this ODE without initial conditions and I get another solution [itex]w[/itex], then it can be regarded as a function of an arbitrary constant: [itex]w=w(C)[/itex].

Is it true to say that [itex]u = w(C)[/itex] for some C? If so, how do I find such a C?
 
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  • #2
That's a little vague. You use the boundary conditions to find C. In general, an ODE doesn't even have a unique solution unless you make some assumptions about the form of the ODE. Can you be more concrete?
 

FAQ: Linear ODE Solutions Without Initial Conditions and the Arbitrary Constant C

What is the Theory of linear ODEs?

The Theory of linear ODEs, also known as the Theory of linear ordinary differential equations, is a mathematical framework for studying and solving differential equations. It deals specifically with linear differential equations, which are equations that involve only the first derivative of the dependent variable and do not contain any higher powers or products of the dependent variable.

What are the key concepts in the Theory of linear ODEs?

The key concepts in the Theory of linear ODEs include the general solution, particular solutions, and the initial value problem. The general solution is the set of all possible solutions to a given differential equation. A particular solution is a specific solution that satisfies the given initial conditions. The initial value problem involves finding a particular solution that satisfies a set of initial conditions.

How is the Theory of linear ODEs used in real-world applications?

The Theory of linear ODEs has many practical applications in fields such as physics, engineering, economics, and biology. It is used to model and analyze a wide range of phenomena, including population growth, heat transfer, electrical circuits, and chemical reactions. In these applications, the theory allows us to make predictions and understand the behavior of complex systems.

What are the main methods for solving linear ODEs?

The main methods for solving linear ODEs include separation of variables, integrating factors, and variation of parameters. Separation of variables involves rewriting the differential equation in a form where the dependent and independent variables are separated. Integrating factors involve multiplying the differential equation by a suitable function to make it easier to solve. Variation of parameters involves finding a particular solution by assuming it has the same form as the general solution.

Are there any limitations to the Theory of linear ODEs?

Yes, the Theory of linear ODEs has limitations. It can only be applied to linear differential equations, which means it cannot be used to solve nonlinear differential equations. Additionally, the theory assumes that the coefficients in the differential equation are constant, which may not always hold true in real-world applications. It is important to carefully consider the assumptions and limitations of the theory when applying it to a problem.

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