Differential equation of dy/dx = (2x-y+4)/ (4x-2y +1)

In summary, the two given solutions to the first order differential equation dy/dx = (2x-y+4)/ (4x-2y +1) differ by a constant summand of 4/9. This difference does not affect the validity of the solutions, as the derivative of a constant is 0.
  • #1
hotjohn
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Homework Statement


I'm asked to use the transformation of v= 2x-y to solve dy/dx = (2x-y+4)/ (4x-2y +1) the answer given is (2/9)(6x-3y-2) +(7/9)ln(6x-3y-2) = x +c , i got (2/9)(6x-3y) +(7/9)ln(6x-3y-2) = x +c , what's wrong with my working ?

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The Attempt at a Solution

 

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  • #2
The two answers differ by a constant summand of 4/9. As you have "+c" anyway, this difference does not matter.
 
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  • #3
mfb said:
The two answers differ by a constant summand of 4/9. As you have "+c" anyway, this difference does not matter.
lol , why the author want to dd in another (2/9)(-2) into the answer ? this is confusing ...
 
  • #4
Presumably, because the author used a different method to arrive at a solution. In any case, the point is that, because the derivative of a constant is 0, two functions, differing by a constant, can be solutions to the same first order differential equation.
 
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Related to Differential equation of dy/dx = (2x-y+4)/ (4x-2y +1)

What is a differential equation?

A differential equation is an equation that involves an unknown function and its derivatives. It describes the relationship between a function and its rate of change.

How do you solve a differential equation?

To solve a differential equation, you need to find the function that satisfies the equation. This can be done through various methods such as separation of variables, substitution, or using the appropriate integrating factor.

What does dy/dx mean in this differential equation?

In this differential equation, dy/dx represents the derivative of the unknown function y with respect to the independent variable x, which is the rate of change of y with respect to x.

What are the initial conditions of this differential equation?

The initial conditions of a differential equation are the values of the unknown function and its derivatives at a specific point or interval. In this equation, the initial conditions are not specified and therefore the solution will be a general solution.

Can this differential equation be solved analytically?

Yes, this differential equation can be solved analytically using the aforementioned methods. However, for more complex equations, numerical methods may be required to find an approximate solution.

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