What is the integral of dydx=y^2/x^2

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In summary, the integral in the equation dy/dx=y^2/x^2 represents the area under the curve of the function with respect to x. It can be solved using various integration techniques and the general steps involve rewriting the function, applying the appropriate technique, integrating, and adding the constant of integration. While some calculators can solve it, it is important to understand the manual steps for accuracy. In real-world applications, the integral is used in physics, engineering, and economics for calculating work, average value, and in economic models.
  • #1
math_trouble
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im having problem integrating the equation

dydx=y^2/x^2

and also

dydx=3*y^2/x
 
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  • #2
I'm assuming they should be dy/dx=y^2/x^2
and dy/dx=3*y^2/x .

Your differential equations textbook should discuss "separation of variables" near the very beginning.
 
  • #3
This can be rewritten in this way:
y'=y^2/x^2 with x different from zero.
y'/y^2=1/x^2
using chain rule:
d/dx[-1/y]=d/dx[-(1/x)+C]
consequentely:
1/y=(1/x)-C
y=1/[(1/x)+C]
 

1. What is the meaning of the integral in dy/dx=y^2/x^2?

The integral in this equation represents the area under the curve of the function y^2/x^2 with respect to the variable x. It is a mathematical operation used to find the total value of a function within a given range.

2. How is the integral of dy/dx=y^2/x^2 solved?

The integral of dy/dx=y^2/x^2 can be solved using various integration techniques such as substitution, integration by parts, or partial fractions. The specific method used will depend on the complexity of the function.

3. What are the steps for solving the integral of dy/dx=y^2/x^2?

The general steps for solving the integral of dy/dx=y^2/x^2 are:

  1. Rewrite the function in the form of y^2/x^2.
  2. Apply the appropriate integration technique.
  3. Integrate the function with respect to x.
  4. Add the constant of integration.

4. Can the integral of dy/dx=y^2/x^2 be solved using a calculator?

Yes, some calculators have built-in integration functions that can be used to solve the integral of dy/dx=y^2/x^2. However, it is important to note that these calculators may not always give the most accurate results and it is still important to understand the steps for solving the integral manually.

5. How is the integral of dy/dx=y^2/x^2 used in real-world applications?

The integral of dy/dx=y^2/x^2 has various applications in physics, engineering, and economics. For example, it can be used to calculate the work done by a variable force or the average value of a function. It is also commonly used in economic models to determine the area under a demand or supply curve.

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