# Science & engineering math: system of differential equations

• chatterbug219
In summary: I would definitely start off as ehild has suggested.There is, however, a problem with the entire exercise. Doing as ehild suggests gives you a second order differential equation in y only which you can solve and then use the initial conditions to give a specific solution for y. But there is no derivative of z in these equations- once you know y, z is fixed and you have no constant to choose to make z(0)= 0. Was one or both of those "z"s supposed to be z'? If not then anyone of the three conditions, y(0)= 3, y'(0)= -2, z(0)=n 0, can be dropped to give a solution but there is not
chatterbug219

## Homework Statement

Solve the system of differential equations:
y'(t) + z(t) = t
y"(t) - z(t) = e-t
Subject to y(0) = 3, y'(0) = -2, and z(0) = 0

## Homework Equations

My professor did an example in class that was much simpler and solved it using Kramer's rule.

## The Attempt at a Solution

I don't know how to start it. I thought about rearranging the equations so that one was equal to y'(t) and the other was equal to z(t), but I'm not sure that would work...

ehild

I would definitely start off as ehild has suggested.

There is, however, a problem with the entire exercise. Doing as ehild suggests gives you a second order differential equation in y only which you can solve and then use the initial conditions to give a specific solution for y. But there is no derivative of z in these equations- once you know y, z is fixed and you have no constant to choose to make z(0)= 0. Was one or both of those "z"s supposed to be z'? If not then anyone of the three conditions, y(0)= 3, y'(0)= -2, z(0)=n 0, can be dropped to give a solution but there is not y, z, satisfying the equations and all three of the conditions.

Oh my gosh yes there was supposed to be a z' in the first equation
So it is: y'(t) + z'(t) = t
The second equation is correct though, so sorry for any confusion!

chatterbug219 said:
Oh my gosh yes there was supposed to be a z' in the first equation
So it is: y'(t) + z'(t) = t
The second equation is correct though, so sorry for any confusion!

Then you can integrate the first equation and add to the second.

ehild

## 1. What is a system of differential equations?

A system of differential equations is a set of equations that describe the relationship between multiple variables and their rates of change over time. It is often used in mathematical modeling to understand complex systems and predict their behavior.

## 2. How is a system of differential equations solved?

There are several methods for solving a system of differential equations, including analytical methods (such as separation of variables and integrating factors) and numerical methods (such as Euler's method and Runge-Kutta methods). The method chosen depends on the complexity of the system and the level of accuracy required.

## 3. What is the importance of using a system of differential equations in science and engineering?

A system of differential equations allows us to mathematically model and understand the behavior of complex systems that cannot be easily studied through traditional experimental methods. It is essential in many fields of science and engineering, including physics, biology, economics, and engineering.

## 4. Can a system of differential equations accurately predict the behavior of a system?

The accuracy of a system of differential equations depends on the accuracy of the model and the methods used to solve it. In some cases, the system may provide a close approximation of the actual behavior of the system, while in other cases, it may not be as accurate. It is important to carefully consider the assumptions and limitations of the model when using a system of differential equations to make predictions.

## 5. Are there any real-world applications of using a system of differential equations?

Yes, there are numerous real-world applications of using a system of differential equations. For example, in physics, it can be used to model the motion of celestial bodies or the behavior of electric circuits. In biology, it can be used to understand population dynamics or the spread of diseases. In engineering, it can be used to design and optimize systems such as engines and bridges.

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