Can This Differential Equation Solution Be Simplified Further?

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In summary, the conversation discusses solving a differential equation using integration and substitution. The final solution is simplified to ln | y + 2 / y - 2 | + C = 4x, with the possibility of further simplification using partial fractions.
  • #1
Naeem
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I did the following:

dy/(y^2 -4 ) = dx

Integral ( -1/4 /y + 2 + 1/4 / ( y-2) . dy = Integral dx

-1/4 ln | y + 2 | + 1 /4 ln |y -2 | + C = x

Multiplying all thru by - 4 we get

ln | y + 2 | + ln | y - 2 | + 4C = 4x ( Note: 4C is another 'C'which is a bigger C)

ln | y + 2 / y - 2| + C = 4x

C = 4x - ln | y + 2 / y -2 |

If this is correct, can we simplify the solution any further.
 
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  • #2
You ahve a lot of mismatched parentheses but,

[tex] \frac{1}{y^2-4} = \frac{1}{(y-2)(y+2)} [/tex]

You can't separate these two in the denominator.

I would do this using the substitution y = 2sec(t)
 
  • #3
Why not? Using Partial fractions you could

A/ something + B / something
 
  • #4
I don't think that's what he did, or if he did he didnt show the work. I can barely read his post.
 
  • #5
That "-4" multiplication is incorreclty made.You should have other signs...

Daniel.
 

FAQ: Can This Differential Equation Solution Be Simplified Further?

What is the equation "Solve : ( y^2 - 4 )dx - dy = 0" used for?

The equation "Solve : ( y^2 - 4 )dx - dy = 0" is commonly used in differential calculus to find the general solution of a first-order differential equation.

How do you solve the equation ( y^2 - 4 )dx - dy = 0?

To solve ( y^2 - 4 )dx - dy = 0, you can use the method of separation of variables, where you separate the variables dx and dy on opposite sides of the equal sign and integrate both sides to find the general solution.

Can you provide an example of solving ( y^2 - 4 )dx - dy = 0?

Yes, for example, if the equation is ( y^2 - 4 )dx - dy = 0, we can separate the variables and integrate to get the general solution y = 2x + C, where C is a constant.

What is the significance of the constant C in the general solution?

The constant C represents the value of the solution at a particular point, and it is determined by the initial conditions of the specific problem being solved.

Can the equation ( y^2 - 4 )dx - dy = 0 be solved using other methods?

Yes, there are other methods such as the method of integrating factors and the method of exact equations that can be used to solve ( y^2 - 4 )dx - dy = 0. However, the method of separation of variables is the most commonly used method for this type of differential equation.

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