Solving a Differential Equation: Exploring Different Methods

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In summary, the two solutions to this differential equation are clearly different. One is based on integrating v by dt, while the other uses the function s/t to solve for t.
  • #1
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I found this exercise in an old physics schoolbook. I maeged to solve it but I don't really understand why I have to solve it in this excact order. Before I start I would just like to add that I have very little experience with differential equations (they are not in our high school year plan), and therefor any explanations will be welcomed.

So here it goes.

t=?
s=10
v=2s+1
v=ds/dt
dt=(1/v)ds
t=Iteg[1/v]ds
t=1/2Ln(2s+1)
This is the way it was solved in the book (I might have memorized the integration of v wrong).

But I tried to solve it like this
t=?
s=10
v=2s+1
v=ds/dt
ds=v*dt
s=(2s+1)t
s/(2s+1)=t

The solutions are clearly different. I don't understend why. The only idea I got was that I can't integrate v by dt becouse t is not a veriable in v and v is therefore threted as a constent what changes the whole thing (is this true). But if there are any beter explanations I would be glad to hear them. Thanks.
 
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  • #2
Neither v nor t is a constant here nor can be treated as one. When you write
"ds= v dt so s= (2s+1)t" you are treating v as a constant- and it is not. It depends upon t. Since you don't yet know v or s as a function of t, the only way to integrate
ds= (2s+1)dt is as ds/(2s+1)= dt which is exactly the book's solution.
 
  • #3
So in general I should always integrate in such a way that the expresion is not threated as a constant, if posibel of course. Right.
 
  • #4
Not just "if possible". You should not assume a variable is a constant when it is not!

What you are doing there is separating the variables- so that you have only s on one side of the equation and t on the other. If that is NOT possible then it is not a "separable" equation and you will have to use some other method. If x is a function of t then you cannot integrate [tex]\int f(x,t)dt[/tex] without knowing exact what function it is!
 
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  • #5
Thank you! Your explanation is really very helpful. :smile:
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives, which represent the rate of change of a function, to model real-world phenomena and solve problems in various fields, such as physics, engineering, and economics.

2. What are the different methods for solving a differential equation?

There are various methods for solving a differential equation, including separation of variables, integrating factors, substitution, and power series. Each method has its own advantages and is suitable for different types of differential equations. It is important to choose the appropriate method based on the given equation and initial conditions.

3. How do I know which method to use for a specific differential equation?

The choice of method depends on the type of differential equation and its initial conditions. For example, if the equation is separable, then the separation of variables method can be used. If the equation is linear, then the integrating factor method can be applied. It is important to first identify the type of equation and then choose the appropriate method accordingly.

4. What are the steps involved in solving a differential equation?

The general steps for solving a differential equation are as follows: 1) Identify the type of equation, 2) Determine the appropriate method, 3) Apply the chosen method to the equation, 4) Solve for the constant of integration, if applicable, 5) Check the solution by substituting it back into the original equation, and 6) Apply the initial conditions, if given, to find the particular solution.

5. Are there any real-world applications of differential equations?

Yes, differential equations are widely used in various fields to model and solve real-world problems. For example, they are used in physics to describe the motion of objects, in engineering to analyze and design systems, in economics to model population growth, and in biology to study the spread of diseases. They provide a powerful tool for understanding and predicting the behavior of complex systems.

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