# A simple differential equation problem.

• MathematicalPhysicist
In summary, the conversation is about solving a physics problem involving an equation with many symbols representing different variables. The experts suggest using an integrating factor to solve the problem, while others mention that the equation can also be solved using separation of variables. The conversation also touches upon the importance of clearly defining the symbols used in the equation.
MathematicalPhysicist
Gold Member
is there a solution to this euqation: (i got to in a problem in physics so the notation is appropiate):
$$(m_1+m_2)\frac{dv}{dt}=m_1g-m_2gsin(a)-f_s-\beta v-V\rho g$$
is there a solution or perhaps i got it wrong in equations?

Well, the notation might be appropriate, but your definition of the symbols is not! :grumpy:
Assuming that everything is constant apart from v,
just rewrite your diff. eq as:
$$\frac{dv}{dt}+Av=B$$
where A and B are constants.
You may use an integrating factor to solve this problem.
This will also work for non-constant A and B (depending solely on t, and not on v!), but you might not get an explicit formula in terms of elementary functions then.

Last edited:
what's wrong with my interpratation of the symbols?

It is lacking. How should I know what you mean by your symbols when you don't define them?

well, the m's are masses, the angle a doesn't change with time (perhaps here lies your critic), V stands for volume and rho stand for water density.
that's it.

perhaps, v=Acos(b)+Bsin(b) is a possible solution here?

Hmm..and I guess that g is the acceleration due to gravity and beta is the coefficient of air resistance. But you didn't sayu that.
Furthermore f_s is some sort of applied force I don't know what is.

what integrating factor to use here?
dv/v(A+B/v)+dt=0 will that suffice here?

this is a pretty simple integrating factor

if you mmove the resistance term (Bv) to the left hand side and divide by (m1+m2) than you have an quation of the formdv/dt +Pv = Q

where Q is some function of t, and so is P

in a problem like this the integrating factor is e^(integral of P dt)

multiplying by that and simplifying will get you an equation of the formd(ve^(integral P dt))/dt =Qe^(integral P dt)

integrate, and then you'll have a simple algebra problem

also notice that the solution only has 1 parameter, all first order differential equation have only one parameter in the solution.

Why use an integrating factor? The ODE's clearly separable...

Why use separation of variables?
You can clearly use an integrating factor..

It's just a matter of preference what method you use.

hmm it doesn't appear to be seperable.

if you note its of the form dv/dt +Bv=C

as somebody else mentioned, that is not a seperable equation

So, you disagree that the diff.eq may be rewritten as:
$$\frac{1}{C-Bv}\frac{dv}{dt}=1$$ ??

oh my mistake, didn't see that manipulation

## 1. What is a differential equation?

A differential equation is a mathematical equation that describes how a dependent variable changes in relation to one or more independent variables. It involves derivatives, which represent the rate of change of the dependent variable.

## 2. What makes a differential equation "simple"?

A simple differential equation is one that can be solved using basic algebraic techniques, without the need for advanced mathematical methods. This typically involves equations with only one independent variable and a limited number of terms.

## 3. How do I solve a simple differential equation?

To solve a simple differential equation, you can use techniques such as separation of variables, substitution, or integrating factors. These methods involve manipulating the equation to isolate the dependent variable and then integrating both sides to find a general solution.

## 4. What are some real-world applications of differential equations?

Differential equations are used to model a wide range of phenomena in various fields, including physics, engineering, economics, and biology. For example, they can be used to describe the growth of populations, the flow of fluids, and the spread of diseases.

## 5. Are there any limitations to using differential equations to model real-world problems?

While differential equations are a powerful tool for modeling complex systems, they do have some limitations. For instance, they may not accurately represent systems that are highly nonlinear or have discontinuous behavior. In these cases, more advanced techniques may be needed to accurately model the system.

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