# Solve DE: Reviewing Problems Before Classes Start

• MHB
• karush
In summary, the equation $e^{5x}\frac{dy}{dx}+ 5e^{5x}y= e^{4x}$ can be solved by dividing through by $e^{5x}$ and solving for $y$ as a linear differential equation with constant coefficients. The general solution is $y(x)= C'e^{-5x}+ \frac{e^{-x}}{4}. karush Gold Member MHB$\tiny [9.1.98]$solve 123$\begin{array}{rl}
e^{5x}\,\frac{dy}{dx} + 5e^{5x}y
&= e^{4x} \\
\dfrac{d}{dx}\left( e^{5x}y\right)
&=e^{4x} \\
e^{5x}y&= \int{ e^{4x}dx} \\
e^{5x}y
&= \frac{1}{4}e^{4x} + C \\
y &= \frac{1}{4}\,{e}^{-x} + C\,{e}^{-5\,x}
\end{array}$just reviewing some problems before Sept classes start up I think this ok not sure how to check it with W|A possible typos Mahalo https://dl.orangedox.com/wr9JnddSXWGHrASWF2 Last edited: Very good! Here is another way to solve that equation (how I would probably have done it myself): The equation is$e^{5x}\frac{dy}{dx}+ 5e^{5x}y= e^{4x}$.​ Divide through by$e^{5x}$to get​$\frac{dy}{dx}+ 5y= e^{-x}$.​ That is a "linear differential equation with constant coefficients". It's homogeneous part is$\frac{dy}{dx}+ 5y= 0$which is the same as$\frac{dy}{dx}= -5y$and then, in "differential" form,$\frac{dy}{y}= -5dx$. Integrating both sides,$ln(y)= -5x+ C$. Taking the expontial of both sides,$y= C'e^{-5x}$where$C'= e^C$is just another constant. That is the general solution to the "associated homogeneous equation".​ Now to find a solution to the entire equation, since "$e^{-x}$is a type of function we get as solution to a "linear differential equation with constant coefficients", we try a solution of the form$y= Ae^{-x}$. Then$y'= -Ae^{-x}$and the equation becomes$-Ae^{-x}+ 5Ae^{-x}= 4Ae^{-x}= e^{-x}$. Since$e^{-x}$is never 0 we can divide by it to get 4A=1 so$A= \frac{1}{4}$.$\frac{e^{-x}}{4}$is a solution to the full equation so the general solution to the full equation is$y(x)= C'e^{-5x}+ \frac{e^{-x}}{4}\$.

Mahalo

## 1. What are differential equations?

Differential equations are mathematical equations that describe how a quantity changes over time, based on its current value and the rate at which it is changing. They are used to model many real-world phenomena, including motion, growth, and decay.

## 2. Why is it important to review differential equation problems before classes start?

Reviewing differential equation problems before classes start allows you to familiarize yourself with the concepts and techniques that will be covered in class. This can help you feel more confident and prepared, and can also help you identify any areas that you may need to focus on.

## 3. What are some common techniques for solving differential equations?

Some common techniques for solving differential equations include separation of variables, substitution, and using an integrating factor. Other techniques, such as Laplace transforms and power series methods, may also be used for more complex problems.

## 4. How can I improve my problem-solving skills for differential equations?

One way to improve your problem-solving skills for differential equations is to practice solving a variety of problems. This can help you become more familiar with different techniques and approaches, and can also help you identify any areas that you may need to work on.

## 5. What resources are available for reviewing differential equation problems?

There are many resources available for reviewing differential equation problems, including textbooks, online tutorials, and practice problem sets. Your professor or TA may also provide review materials or hold review sessions before exams. Additionally, working with a study group or seeking help from a tutor can also be beneficial.

• Differential Equations
Replies
6
Views
1K
• Differential Equations
Replies
1
Views
953
• Differential Equations
Replies
2
Views
1K
• Differential Equations
Replies
10
Views
1K
• Differential Equations
Replies
4
Views
987
• Differential Equations
Replies
2
Views
818
• Differential Equations
Replies
2
Views
2K
• Differential Equations
Replies
2
Views
2K
• Differential Equations
Replies
7
Views
2K
• Differential Equations
Replies
4
Views
1K