# Solve DE for y: $\displaystyle y^\prime +y = xe^{-x}+1$

• MHB
• karush
In summary, the given differential equation was solved by dividing every term by the integrating factor, simplifying and reordering terms to get the final answer of $y(x)=c_1e^{-x}+\frac{1}{2}e^{-x}x^2+1$. The steps taken include integrating, multiplying by the integrating factor, and simplifying the resulting expression.
karush
Gold Member
MHB
$\textsf{Given:}$
$$\displaystyle y^\prime +y = xe^{-x}+1$$
$\textit{Solve the given differential equation}$
$\textit{From:$\displaystyle\frac{dy}{dx}+Py=Q$}$
$\textit{then:}$
$$\displaystyle e^x y=\int x+e^{-2x} \, dx + c \\ \displaystyle e^x y=\frac{1}{2}(x^2-e^{-2x})+c$$
$\textit{divide every term by$e^x$}$
$$\displaystyle y=\frac{1}{2(e^x)}(x^2) -\frac{e^{-2x}}{2(e^x)}+\frac{c}{(e^x)}$$
$\textit{simplify and reorder terms}$
$$\displaystyle y=c_1e^{-x}+\frac{1}{2}e^{-x}x^2+1$$
$\textit{Answer by W|A}$
$$y(x)=\color{red} {\displaystyle c_1e^{-x}+\frac{1}{2}e^{-x}x^2+1}$$

any bugs any suggest?

The integrating factor is:

$$\displaystyle \mu(x)=e^x$$

And so we get:

$$\displaystyle y'e^x+ye^x=x+e^x$$

$$\displaystyle \frac{d}{dx}\left(e^xy\right)=x+e^x$$

Integrate:

$$\displaystyle e^xy=\frac{x^2}{2}+e^x+c_1$$

Hence:

$$\displaystyle y(x)=\frac{x^2}{2}e^{-x}+1+c_1e^{-x}$$

Somehow you arrived at the correct answer, but your work doesn't show how.

MarkFL said:
Hence:

$$\displaystyle y(x)=\frac{x^2}{2}e^{-x}+1+c_1e^{-x}$$

Somehow you arrived at the correct answer, but your work doesn't show how.

ok I hit and missed with some examples
but see your steps make a lot more sense

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates one or more functions and their derivatives. It is used to describe the relationship between a function and its rate of change.

## 2. How do you solve a differential equation?

To solve a differential equation, you must find a function that satisfies the equation. This involves using various techniques such as separation of variables, integration, and substitution.

## 3. What is the solution to the given differential equation?

The solution to the given differential equation is y = (x+2)e-x. This can be found by using the method of integrating factors.

## 4. Can you explain the steps taken to solve this particular differential equation?

To solve this differential equation, we first rearrange it to the form y' + Py = Q, where P and Q are functions of x. In this case, P=1 and Q=xe-x+1. Then, we find the integrating factor e∫Pdx, which in this case is ex. Multiplying both sides of the equation by this integrating factor, we get exy' + yex = x+2. This can then be solved by integrating both sides and applying the initial condition to find the value of the constant.

## 5. How are differential equations used in real life?

Differential equations are used in a variety of fields such as physics, engineering, economics, and biology to model and understand real-world phenomena. They are particularly useful in predicting how a system will change over time and are essential in the development of many technologies and processes.

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