# Differential Equations Variation of Params Problems

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In summary, the conversation discusses solving a differential equation with the given parameters using the Variation of Parameters formula and the Wronskian determinant. The attempt at solution involves finding the Wronskian and using it to solve for the coefficients u and v. However, the integral method used by the speaker gives a different answer than the correct solution. The speaker is unsure of where they went wrong and would appreciate any guidance in finding the error.

## Homework Statement

$$t^{2}$$ * y'' - 2y = $$3t^{2}$$ - 1

$$y_{1}$$ = t$$^{2}$$

$$y_{2}$$ = 1/t

## Homework Equations

Variation of Params forumla
wronskian det

## The Attempt at a Solution

W = t^2 * -t^-2 -[1/t * 2t] = -3

Y = -t^2 * Integral[ (-1/3)(1/t)(3t^2 -1)] + 1/t * Integral[(-1/3)(t^2)(3t^2 -1)]

I get an answer that is not correct so I guess I set this part up wrong somehow anyone see the error?

My answer: Y = 3/10*t^4 - t^2 * ln|t|/3

Correct Answer: Y = t^2*ln|t| + 1/2

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So I figured I ended up getting the correct answer but going the long way and solving the system using

u*t^2 + v * 1/t = 3t^2 -1

but I still have no idea why the integral gave me the wrong answer apparently I missed something somewhere, if you feel like going through please let me know what I messed up on.

## 1. What are "Differential Equations Variation of Params Problems"?

"Differential Equations Variation of Params Problems" are a type of mathematical problem that involves solving a differential equation by varying some of the parameters in the equation. This method is often used to solve nonhomogeneous differential equations, where the coefficients are functions of the independent variable.

## 2. What is the purpose of solving these types of problems?

The purpose of solving "Differential Equations Variation of Params Problems" is to find a particular solution to a nonhomogeneous differential equation. This can be useful in many scientific and engineering applications, such as modeling the behavior of physical systems.

## 3. What are the steps involved in solving a "Differential Equations Variation of Params Problem"?

The first step is to rewrite the differential equation in standard form, by dividing through by the coefficient of the highest order derivative. Next, we find the complementary solution by solving the associated homogeneous equation. Then, we use the variation of parameters method to find the particular solution, by assuming a general form for the particular solution and solving for the parameters. Finally, we combine the complementary and particular solutions to obtain the general solution.

## 4. What are some common applications of "Differential Equations Variation of Params Problems"?

"Differential Equations Variation of Params Problems" have many applications in science and engineering, such as in circuit analysis, population dynamics, and chemical reactions. They are also used in various fields of physics, including mechanics, electromagnetism, and thermodynamics.

## 5. Are there any limitations or challenges in solving "Differential Equations Variation of Params Problems"?

One limitation is that this method can only be used for nonhomogeneous differential equations, so it may not be applicable to all types of problems. Additionally, finding the particular solution can be challenging, as it involves solving integrals and working with potentially complex functions. It also requires a good understanding of the associated homogeneous equation and the method of undetermined coefficients.