General solution for given y'(x) = 2/(7-4y)

In summary, the conversation is about a question for homework that involves finding the general solution (y=...) for the equation y'(x) = 2/(7-4y). The conversation discusses using the method of separating variables and integrating to find the solution, and also mentions the use of the quadratic formula. The person asking for help is grateful for the explanation and compliments the forum for being very responsive.
  • #1
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could some1 help me with the working of this question. it's for homework and i would help if i knew how to come to the answer in the back of the book.

it says to find the general solution (y=...)

given y'(x) = 2/(7-4y)

...


how do u get to this:
y = 7/4 +/- √(k-x)

that is: seven over four plus or minus the squareroot of k minus x

k being another constant

please help if u can

thanx
 
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  • #2
Surely your textbook (not to mention your teacher!) has talked about "separating variables"!

y'= 2/(7- 4y) is the same as [itex]\frac{dy}{dx}= \frac{2}{7-4y}[/itex]

"Separate" that into differential form with y on one side of the equation and x on the other:
(7- 4y)dy= 2dx.
Integrating: 7y-2y2= 2x+ k.

Now think of that as the quadratic equation 2y2- 7y+ (2x+k)= 0 and solve for y using the quadratic formula. (The "C" in Ay2+ By+ C is 2x+k.)
 
  • #3
thank you very much

this is a great forum

very responsive
 

1. What is a general solution?

A general solution is a mathematical expression that contains all possible solutions to a given differential equation. It is represented by using a constant, usually denoted as C, which can take on different values to yield different specific solutions.

2. How do you find the general solution for a given differential equation?

To find the general solution, you need to integrate the given differential equation using standard integration techniques. This will result in a general expression containing the dependent variable and the constant C. The constant C can be determined by using initial conditions or boundary conditions.

3. What is the difference between a particular solution and a general solution?

A particular solution is a specific solution to a given differential equation that satisfies certain initial or boundary conditions. On the other hand, a general solution contains all possible solutions to a differential equation and is represented by a constant C, which can take on different values to yield different particular solutions.

4. Can a general solution be used to solve any differential equation?

Yes, a general solution can be used to solve any differential equation. However, the constant C must be determined by using initial or boundary conditions specific to the problem at hand.

5. How can a general solution be verified?

A general solution can be verified by plugging it back into the original differential equation. If the general solution satisfies the differential equation, it is a valid solution. Additionally, any specific solution obtained by substituting a value for the constant C should also satisfy the differential equation.

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