# Differential equations question

• DWill
In summary, differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model various phenomena in science and engineering and the purpose of solving them is to find a function that accurately models the behavior of a system. There are different types of differential equations, including ordinary, partial, and stochastic, each with their own characteristics and applications. These equations can be solved analytically or numerically, and they have wide-ranging applications in fields such as physics, engineering, biology, economics, and finance.
DWill

## Homework Statement

Let y = y1(t) be a solution of
y' + p(t)y = 0, (i)

and let y = y2(t) be a solution of
y' + p(t)y = g(t). (ii)

Show that y = y1(t) + y2(t) is also a solution of Eq. (ii)

## The Attempt at a Solution

I'm not really sure how to start this one. Do I try to solve both (i) and (ii) first?

No. Just substitute y=y1(t)+y2(t) into the second equation and use the first one to simplify it.

## 1. What are differential equations?

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model various phenomena in science and engineering, such as growth, decay, and motion.

## 2. What is the purpose of solving differential equations?

The purpose of solving differential equations is to find the function that satisfies the equation and accurately models the behavior of a system. This allows scientists to make predictions and understand the underlying mechanisms of a phenomenon.

## 3. What are the different types of differential equations?

There are several types of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve only one independent variable, while PDEs involve multiple variables. SDEs also involve random variables and are commonly used in finance and physics.

## 4. How are differential equations solved?

Differential equations can be solved analytically, using mathematical techniques such as separation of variables and integrating factors. They can also be solved numerically, using computer algorithms to approximate the solution.

## 5. What are some applications of differential equations?

Differential equations have a wide range of applications in various fields, including physics, engineering, biology, economics, and finance. They are used to model and predict the behavior of systems such as population growth, chemical reactions, and electrical circuits.

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