Derivation of Variation of Paramters

In summary, the variation of parameters method is used in ordinary differential equations to solve linear second order equations. The particular solution is of the form yp = v1y1 + v2y2, and the derivative with respect to t is taken to yield an equation with two unknown functions, v1 and v2. To simplify computation and avoid second-order derivatives, the requirement is imposed that y1*dv1/dt + y2*dv2/dt = 0. This second equation was identified by Lagrange and allows for solving for v1 and v2.
  • #1
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First time poster, so please feel free to leave any comments of a general nature.

I'm hoping to get a further insight on the derivation of the variation of parameters method used in ordinary differential equations to solve linear second order equations. I understand were looking for a particular solution of the form
[tex]
y_{p}=v_{1}(t) \cdot y_1(t)+v_2(t) \cdot y_2(t)
[/tex]
Taking the derivative with respect to t yields
[tex]
\dfrac{d}{dt}y_p=(y_1 \cdot \dfrac{d}{dt}v_1 + y_2 \cdot \dfrac{d}{dt}v_2) + (v_1 \cdot \dfrac{d}{dt}y_1 + v_2 \cdot \dfrac{d}{dt}y_2)
[/tex]
The next step is to impose the requirement that
[tex]
y_1 \cdot \dfrac{d}{dt}v_1 + y_2 \cdot \dfrac{d}{dt} v_2 = 0
[/tex]

It is at this point where I am not understanding (actually, this is the only step I don't understand in the derivation) . While I see that having the second derivatives of the functions [tex] v_1 [/itex] and [tex] v_2 [/itex] is going to be problematic, I do not understand why this requirement can be imposed without further explanation beyond "To simplify computation and to avoid second-order derivatives for the unknowns v1 and v2in the expression y''_p, we impose the requirement" (looked at a couple derivations online and in my old textbook, quote is from my textbook right before they impose the requirement I don't understand). Perhaps it is beyond the scope of the course, or perhaps I'm overlooking something blatantly simple? Any insight is appreciated.
 
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  • #2
Hi blinktx411, Welcome to the Differential Equations forum.

Remember that you are looking for a particular solution of the form yp=v1y1 + v2y2 .
You have two unknown functions v1 and v2 to be determined. But how many known equations do you have to solve this? Only one, from the given DE.

So you are free to do with what you like for the second equation as long as you are able to solve for v1 and v2 . (remember that you are just looking for a particular solution).

It took the genius of Lagrange to identified that second equation. And the method also works nicely with higher order.
 

1. What is the derivation of variation of parameters?

The derivation of variation of parameters is a method used in mathematics and physics to solve differential equations. It involves finding a particular solution to a non-homogeneous differential equation by using a linear combination of solutions to the associated homogeneous equation.

2. Why is variation of parameters important?

Variation of parameters is important because it allows us to find the general solution to a non-homogeneous differential equation. This method is particularly useful when the non-homogeneous term in the equation is complex and cannot be solved using other techniques such as the method of undetermined coefficients.

3. How is variation of parameters applied in real-world situations?

Variation of parameters is commonly used in physics and engineering to model real-world systems. For example, it can be used to model the motion of a mass on a spring or the current in an electrical circuit. It can also be applied in various fields such as biology, economics, and finance to study systems that involve changing parameters over time.

4. What are the limitations of variation of parameters?

One limitation of variation of parameters is that it can only be applied to linear differential equations. It also requires the homogeneous solution to be known, which may not always be the case. Additionally, the calculations involved in this method can be quite complex and time-consuming.

5. Are there any alternatives to variation of parameters?

Yes, there are alternative methods for solving non-homogeneous differential equations such as the method of undetermined coefficients and the Laplace transform. These methods may be simpler and more efficient in certain cases, but they also have their own limitations. Therefore, it is important to understand and be familiar with multiple methods for solving differential equations.

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