Solve Differential Equation Using Variation of Parameters

In summary, the problem is to solve y''+25y=10sec(5t). The attempt at a solution was to find the particular solution, which was determined to be 2/5log(cos(5t))cos(5t)+2tsin(5t). However, this answer was initially deemed incorrect, but was later corrected to include an absolute value within the log as it came from the integral of the tangent.
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Homework Statement



Solve y''+25y=10sec(5t)

Homework Equations



NA

The Attempt at a Solution



I believe I have the correct answer for yp which is:

2/5log(cos(5t))cos(5t)+2tsin(5t)

When I plug this into the Webwork field, it says it is incorrect. I checked my answer against Wolfram Alpha and they look the same.

I was hoping someone could look at it and identify if it was actually correct or not. If not, I can go through the steps here and try to work it out.

Thanks
 
Last edited:
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  • #2
I figured it out. The value within the log should be an absolute value as it came from the integral of the tangent.
 

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 that it is a linear combination of the solutions to the corresponding homogeneous equation. This allows us to find a solution that satisfies both the differential equation and any given initial conditions.

2. When should the variation of parameters method be used?

The variation of parameters method is most useful when the non-homogeneous term of the differential equation is a simple function, such as a polynomial or exponential. It can also be used when the homogeneous solution is known, making it easier to find the particular solution.

3. What are the basic steps to solve a differential equation using variation of parameters?

The basic steps to solve a differential equation using variation of parameters are:1. Find the general solution to the corresponding homogeneous equation.2. Assume a particular solution in the form of a linear combination of the solutions from step 1.3. Substitute the particular solution into the original differential equation and solve for the coefficients.4. Add the particular solution to the general solution to obtain the complete solution.5. If initial conditions are given, use them to find the specific solution.

4. Can the variation of parameters method be used for all types of differential equations?

No, the variation of parameters method can only be used for non-homogeneous linear differential equations. It cannot be applied to non-linear or higher order differential equations.

5. How does the variation of parameters method compare to other methods for solving differential equations?

The variation of parameters method is a versatile technique that can be used to solve a wide range of non-homogeneous linear differential equations. However, it can be more complex and time-consuming compared to other methods, such as the method of undetermined coefficients or the Laplace transform method. The choice of method will depend on the specific equation and initial conditions given.

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