Differential Equations: Variation of Parameters

In summary, the conversation discusses finding the particular solution to a differential equation using the method of variation of parameters. The attempted solution involves setting the equation equal to zero and finding the homogeneous solution, then using the method of undetermined coefficients to find the correct answer. The individual eventually figures out their mistake and solves the problem.
  • #1
broodfusion
3
0

Homework Statement


Find the particular solution to the differential equation using method of variation of parameters:

4y''-4y'+y=16e^(t/2)


The Attempt at a Solution



Set 4y''-4y'+y=0

then the homogeneous solution is:

y= c1*e^(t/2)+c2*te(t/2)

set y1= e^(t/2), y2= te^(t/2)

then y1' = (1/2)*e^(t/2), y2' = (t/2+1)*e^(t/2)

Wronskian = W(y1,y2) = e^t

http://img140.imageshack.us/img140/1822/dif1.jpg


I know i did something wrong because checking my answer by plugging Y(t) back into the O.D.E , left hand side and right hand side don't check out.

By using method of undetermined coefficients, Y(t) = 2t^2*e^(t/2), which is the correct answer.

So question is what did i do wrong using method of variation of parameters?

Thanks
 
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  • #2
nevermind, i figured it out myself
 

FAQ: Differential Equations: Variation of Parameters

1. What is the variation of parameters method for solving differential equations?

The variation of parameters method is a technique used to solve non-homogeneous linear differential equations. It involves finding a particular solution by assuming it has the same form as the non-homogeneous term of the equation and then using undetermined coefficients to determine the solution.

2. How is the variation of parameters method different from other methods for solving differential equations?

The variation of parameters method is different from other methods, such as the method of undetermined coefficients or the method of integrating factors, because it does not require the differential equation to be in a specific form. It can be used to solve any non-homogeneous linear differential equation, whereas other methods are limited to certain types of equations.

3. Can the variation of parameters method be used to solve higher-order differential equations?

Yes, the variation of parameters method can be used to solve higher-order differential equations, as long as they are non-homogeneous and linear. The process is the same as for first-order equations, except that multiple derivatives will need to be taken into account when determining the particular solution.

4. Are there any limitations to using the variation of parameters method?

The variation of parameters method may not be the most efficient method for solving certain types of differential equations, such as those with complicated non-homogeneous terms. In these cases, other methods may be more suitable. Additionally, the method may not work if the non-homogeneous term has a special form, such as trigonometric functions.

5. Can the variation of parameters method be used to solve systems of differential equations?

Yes, the variation of parameters method can be extended to solve systems of differential equations. The process involves finding the fundamental set of solutions for the homogeneous system and then using them to determine the particular solutions for the non-homogeneous system. This method is particularly useful for solving coupled systems of equations.

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