# Few questions about solving DE

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In summary, you are correct in your thoughts about solving these differential equations and your particular solution guesses are also correct. It is always a good idea to check your answer using different methods to ensure accuracy.
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## Homework Statement

See the image here : http://gyazo.com/de17b74fd351f26e2e361ae5d975b390

## The Attempt at a Solution

I'm a bit rusty at solving DEs since I was quite far ahead of the rest of the class and haven't practiced much. I'm just wondering if my thoughts about these are correct.

For the first one, I should solve the homogeneous system for a solution yc and then use the method of undetermined coefficients to solve the non-homogeneous one. I believe my particular solution guess should be Y = Ax + b.

Same process for the second one, except my guess should be Y = Asin(2t) + Bcos(2t) + Ce-4t.

For the third one, I'm pretty sure I solve the homogeneous system and then use the method of variation of parameters... or I could use something I figured out on my own time I'm pretty sure. Couldn't I just figure out a particular solution here by using :

$Y(t) = \sum_{m=1}^{n} y_m(t) \int \frac{g(t) W_m(t)}{W(t)}dt$ where ym(t) is one of our known solutions to the homogeneous system, g(t) is our equation on the right hand side of the DE, Wm(t) is the determinant of W obtained by replacing the mth column by (0, 0, 0, ..., 1) ( Cramer's Rule ), and W(t) is the wronskian of my known solutions to the homogeneous equation.

The last one looks ugly. Even though I can solve it using undetermined coefficients, I can tell it's going to get quite messy because if I do, my guess will be :

Y = (At5 + Bt4 + Ct3 + Dt2 + Et + F)e-t

Wouldn't abusing my formula from above again be more desirable here as it would probably reduce the algebra involved?

Thanks for the help in advance.

Last edited:

Hello,

Your thoughts about solving these differential equations are generally correct. For the first one, you are correct in solving the homogeneous system for a solution yc and then using the method of undetermined coefficients to solve the non-homogeneous one. Your particular solution guess, Y = Ax + b, is also correct.

For the second one, your guess of Y = Asin(2t) + Bcos(2t) + Ce^-4t is also correct. This is because the non-homogeneous part of the equation is a sinusoidal function and the complementary solution is also a sinusoidal function. Therefore, your particular solution guess should also be a sinusoidal function.

For the third one, you can use the method of variation of parameters or your formula to find a particular solution. Both methods will give the same result.

Finally, for the last one, you are correct in using the method of undetermined coefficients. Your guess of Y = (At^5 + Bt^4 + Ct^3 + Dt^2 + Et + F)e^-t is correct. While it may seem messy, using the formula you mentioned may not necessarily make it easier. It is always a good idea to check your answer using both methods to ensure accuracy.

I hope this helps. Let me know if you have any further questions.

## 1. What is a differential equation (DE)?

A differential equation (DE) is a mathematical equation that relates a function to its derivatives. It represents a relationship between a variable and its rate of change. Differential equations are used to describe many natural phenomena and are an essential tool in scientific research.

## 2. What is the difference between an ordinary differential equation (ODE) and a partial differential equation (PDE)?

An ordinary differential equation (ODE) involves only one independent variable, while a partial differential equation (PDE) involves two or more independent variables. ODEs are used to model systems where the variables change over a single dimension, while PDEs are used to model systems where the variables change over multiple dimensions.

## 3. What techniques are commonly used to solve differential equations?

There are several techniques used to solve differential equations, including separation of variables, substitution, and integrating factors. Other methods include power series, Laplace transforms, and numerical methods such as Euler's method and Runge-Kutta method.

## 4. Why are initial conditions and boundary conditions important when solving differential equations?

Initial conditions and boundary conditions are necessary when solving differential equations because they provide the starting point and constraints for finding a specific solution. Initial conditions specify the solution at a specific point, while boundary conditions specify the solution at the boundaries of the domain.

## 5. How are differential equations used in real-world applications?

Differential equations have many real-world applications in various fields, including physics, engineering, biology, economics, and more. They are used to model and predict the behavior of systems such as population growth, chemical reactions, and electrical circuits. They are also essential in developing mathematical models for understanding complex systems and making predictions.

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