# Variation of Parameters for System of 1st order ODE

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• hotvette
In summary, Kreyszig Advanced Engineering Mathematics explains the variation of parameter method for a system of first order ODE, where the particular solution is expressed as the product of the homogeneous solution and a function u(x). For a single 2nd order non-homogeneous ODE, the solution involves definite integrals of the homogeneous solutions and the non-homogeneous term. The use of definite integrals may be related to the presence of initial conditions in the problem.
hotvette
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Kreyszig Advanced Engineering Mathematics shows the variation of parameter method for a system of first order ODE: $$\underline{y}' = \underline{A}(x)\underline{y} + \underline{g}(x)$$ The particular solution is: $$\underline{y}_p = \underline{Y}(x)\underline{u}(x)$$ where $\underline{Y}(x)$ is the homogeneous solution and: $$\underline{u}(x) = \int_0^x \underline{Y}^{-1}\underline{g}$$ But, for for a single 2nd order non-homogeneous ODE $y'' + p(x) y' + q(x) y = r(x)$ the solution is:$$y_p = y_1 \int \frac{y_2}{W}r(x)dx + y_2 \int \frac{y_1}{W}r(x)dx$$ where $y_1, \, y_2$ are the solutions to the homogeneous equation and $W$ is the Wronskian of the homogeneous solutions.

Why the definite integral in the case of systems but indefinite integral in the case of a single 2nd order ODE? The book offers no explanation.

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I checked my undergrad math book and it uses indefinite integral for systems. To (hopeful) gain some insight, I worked a couple of problems (one from Kreyszig and one I converted to a system from a single 2nd order ODE) and in both cases, the lower limit of integration produced extraneous solutions that could be absorbed in the homogeneous solution, which suggests the definite integral is worthless. But, through some web searching I found an example using definite integral for initial value problems. Thus, it appears the definite integral is useful if the stated problem has initial conditions. I wonder if the same is true for variation of parameters for non-systems?

## 1. What is the Variation of Parameters method for solving systems of 1st order ODEs?

The Variation of Parameters method is a technique used to solve systems of first order ordinary differential equations (ODEs) that cannot be solved using traditional methods such as separation of variables or substitution. It involves finding a particular solution by varying the coefficients of the general solution of the homogeneous equation.

## 2. When is the Variation of Parameters method typically used?

The Variation of Parameters method is typically used when dealing with non-homogeneous systems of first order ODEs, where the coefficients are functions of the independent variable. It is also useful when the initial conditions for the system are not easily obtainable.

## 3. How does the Variation of Parameters method work?

The Variation of Parameters method involves finding a particular solution by assuming that the coefficients of the general solution of the homogeneous equation are functions of the independent variable. These functions are then substituted into the original system of equations, and the resulting system is solved for the unknown functions. The general solution is then the sum of the particular solution and the general solution of the homogeneous equation.

## 4. What are the advantages of using the Variation of Parameters method?

The Variation of Parameters method allows for the solution of non-homogeneous systems of first order ODEs, which cannot be solved using other methods. It also does not require the initial conditions for the system, making it useful in situations where those conditions are unknown or difficult to obtain.

## 5. Are there any limitations to the Variation of Parameters method?

The Variation of Parameters method can become quite complex and tedious when dealing with higher order systems or systems with more variables. In these cases, it may be more efficient to use other methods such as Laplace transforms or numerical techniques. Additionally, the method may not always yield a closed-form solution and may require further manipulation to obtain a useful solution.

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