Solving ODE Problems: Understanding Tangent Lines and Integrating Functions

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In summary: So when they say y = y' x + b, they are defining a specific tangent line for a specific point on the curve. It is not the equation of the entire family of curves, but just one particular tangent line at one particular point on one particular curve.
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c.teixeira
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I have been reading Ordinary Differential Equations (Pollard) from Dover.
The chapter I am in, is called Problems Leading to Differential Equations of The First Order - Geometric Problems.

Problem :

Find the family of curves with the property that the area of the region bounded by the x-axis , the tangent line drawn at a point P(x,y) of a curve of the family and the projection of the tangent line on the x-axis has a constante value A.

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In the solution, they say the equation of the tangent line is y / (x - a) = y'

They then solve, for a:

a = x - (y/y')

Afterwards, they obtain the distance QR = y/y'

Therefore they have the area of the triangle. They integrate, bla blabla.

Now, when I first looked this, it seemed pretty simple and straighforward. I understood every step. It was an elementary problem.

But, today I gave it a second look, and now I just don't agree with the solution.
---------------
Well, my question is y = mx + b;
but m = y'.

so, y = y' x + b.
I don't agree with this since y defines the equation of the tangent line BUT y' defines the derivative of THE CURVE. therefore in my viewing, when they, in the solution, reach to QR = y/y', and then integrate they are mixing a fuction and a derivative of a diferent fuction.

So, where is my reasoning wrong?
Perhaps I should sleep more. ;D

Thanks for all the explanations!
 
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  • #2
c.teixeira said:
Well, my question is y = mx + b;
but m = y'.

so, y = y' x + b.
I don't agree with this since y defines the equation of the tangent line BUT y' defines the derivative of THE CURVE. therefore in my viewing, when they, in the solution, reach to QR = y/y', and then integrate they are mixing a fuction and a derivative of a diferent fuction.

So, where is my reasoning wrong?
Perhaps I should sleep more. ;D

Thanks for all the explanations!

The slope of a line tangent to a function at a point is the same as the value of the derivative of the function at that point, by definition; this also means that the derivative of the tangent line at a point is the same as the derivative of the function at that point, so [itex]y'_{line} = y'_{curve}[/itex].

Since the line given by [itex]y = mx + b[/itex] is defined to be the tangent line to the curve, that means that [itex]m[/itex] must be equal to the [itex]y'[/itex] of the curve it is tangent to in order to statisfy that condition, which again, happens to also be the [itex]y'[/itex] of the line itself..
 
  • #3
c.teixeira said:
I have been reading Ordinary Differential Equations (Pollard) from Dover.
The chapter I am in, is called Problems Leading to Differential Equations of The First Order - Geometric Problems.

Problem :

Find the family of curves with the property that the area of the region bounded by the x-axis , the tangent line drawn at a point P(x,y) of a curve of the family and the projection of the tangent line on the x-axis has a constante value A.

In the solution, they say the equation of the tangent line is y / (x - a) = y'

They then solve, for a:

a = x - (y/y')

Afterwards, they obtain the distance QR = y/y'

Therefore they have the area of the triangle. They integrate, bla blabla.

Now, when I first looked this, it seemed pretty simple and straighforward. I understood every step. It was an elementary problem.

But, today I gave it a second look, and now I just don't agree with the solution.
---------------
Well, my question is y = mx + b;
Well, it should be y= m(x- a)+ b.

but m = y'.

so, y = y' x + b.
so y= y'(a)(x- a)+ b

I don't agree with this since y defines the equation of the tangent line BUT y' defines the derivative of THE CURVE. therefore in my viewing, when they, in the solution, reach to QR = y/y', and then integrate they are mixing a fuction and a derivative of a diferent fuction.

So, where is my reasoning wrong?
Perhaps I should sleep more. ;D

Thanks for all the explanations!
One definition of "derivative" (at a given point) is "slope of the tangent line" (at that point).
 

1. What is an ODE problem?

An ODE (ordinary differential equation) problem is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model physical phenomena in science and engineering.

2. Are ODE problems difficult?

ODE problems can be difficult, but it depends on the specific problem and your level of understanding and experience with solving differential equations. With practice and the right tools, ODE problems can be solved successfully.

3. How do I solve an ODE problem?

There are various methods for solving ODE problems, such as analytical methods, numerical methods, and computer algorithms. The method used will depend on the complexity of the problem and the desired level of accuracy.

4. Can I use software to solve ODE problems?

Yes, there are many software programs and packages available that can solve ODE problems. Some popular options include MATLAB, Mathematica, and Python libraries like SciPy and SymPy. These tools can provide solutions to ODE problems with high accuracy and efficiency.

5. What are some real-world applications of ODE problems?

ODE problems are used in various fields, such as physics, chemistry, biology, engineering, and economics. They can be used to model and predict the behavior of systems such as population growth, chemical reactions, pendulum motion, and electrical circuits. ODE problems are also essential in understanding and solving real-world problems in these fields.

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