# I'm starting to learn about differential equation

• cambo86
In summary, the conversation is about verifying the general solution of a given differential equation. The method used is to differentiate the solution and substitute it into the original equation to see if it satisfies the DE. The conversation also includes a suggestion to use integrating factors to solve the DE. The final step for verification is to manipulate the differentiated solution to form the general solution again.
cambo86

## Homework Statement

Verify that the differential equation,
${\frac{dy}{dx}} = 15 - {\frac{2y}{81+3x}}$
has the general solution
$y(x) = 3(81+3x) + C(81+3x)^{-2/3}$

2. The attempt at a solution
I've just learned about differential equations, so I'm probably missing something very basic. I've tried serperating x and y so that I can integrate and it's not an exact equation.

Thanks in advance for the help.

cambo86 said:

## Homework Statement

Verify that the differential equation,
${\frac{dy}{dx}} = 15 - {\frac{2y}{81+3x}}$
has the general solution
$y(x) = 3(81+3x) + C(81+3x)^{-2/3}$

2. The attempt at a solution
I've just learned about differential equations, so I'm probably missing something very basic. I've tried serperating x and y so that I can integrate and it's not an exact equation.

Thanks in advance for the help.

Can you differentiate the solution and plug it back into the original differential equation and see if it satisfies the DE?

-Matt

If you want to solve the differential equation, bring the fractional term on the RHS over to the LHS and you see you have a form that can be solved via integrating factors (the resulting eqn is linear in the independent variable y so this method is valid).

hi cambo86!
cambo86 said:
Verify …

"verify" always means that you don't have to solve it, you just assume the answer, and check it!

(by plugging it in, as Matt (leright) says)

So I differentiated the general solution and then subsituted that into the DE. Then I manipulated that to form the general solution again. Is that all that is needed to verify?

cambo86 said:
So I differentiated the general solution and then subsituted that into the DE. Then I manipulated that to form the general solution again. Is that all that is needed to verify?

I don't understand what you are trying to say. You have a function
$$y = y(x) = 3(81+3x)+C(81+3x)^{−2/3}.$$
You can compute dy/dx. When you do that, can you re-write dy/dx as
$$15− \frac{2y}{81+3x}?$$
If your answer is YES, then you have verified the solution. What other possible meaning could the word "verify" have?

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is used to model various physical phenomena and is an important tool in many scientific fields such as physics, engineering, and economics.

## 2. Why are differential equations important?

Differential equations are important because they allow us to describe and understand the behavior of complex systems and processes. They are used to make predictions and solve real-world problems in a variety of disciplines.

## 3. What are some common applications of differential equations?

Differential equations are used in many areas of science and engineering, such as modeling population growth, predicting the motion of objects, analyzing chemical reactions, and understanding electrical circuits. They are also used in economics, finance, and biology.

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

There are several types of differential equations, including ordinary differential equations (ODEs), which involve a single independent variable, and partial differential equations (PDEs), which involve multiple independent variables. Other types include linear and nonlinear differential equations.

## 5. How do I solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some methods include separation of variables, substitution, and using numerical techniques. It is important to have a strong understanding of algebra, calculus, and other mathematical concepts to effectively solve differential equations.

• Calculus and Beyond Homework Help
Replies
25
Views
771
• Calculus and Beyond Homework Help
Replies
3
Views
525
• Calculus and Beyond Homework Help
Replies
7
Views
636
• Calculus and Beyond Homework Help
Replies
27
Views
668
• Calculus and Beyond Homework Help
Replies
10
Views
692
• Calculus and Beyond Homework Help
Replies
10
Views
780
• Calculus and Beyond Homework Help
Replies
7
Views
906
• Calculus and Beyond Homework Help
Replies
24
Views
2K
• Calculus and Beyond Homework Help
Replies
18
Views
2K
• Calculus and Beyond Homework Help
Replies
3
Views
1K