Proof of Solutions for y' = xg(x,y) Equation

• Fibonacci88
In summary, we are given the equation y'= xg(x,y) and assume that g and (partial) dg/dy are defined and continuous for all (x,y). We need to show the following:1) y(x)=0 is a solution This statement is not true in general. A counterexample would be if g(x,y)= 1, then y(x)= (1/2)x^2 + C is a solution. But y(x)= 0 is not a solution since y'= 0 and y'= 0 is not equal to x.2) If y=y(x) is a solution for x in (a,b) and y(x0)>0 for some x0 in (a,b),
Fibonacci88
Given the equation y'= xg(x,y) , suppose that g and (partial) dg/dy are defined and continuous for all (x,y). Show the following:

1) y(x)=0 is a solution

2)if y=y(x), x in (a,b) is a solution and if y(x0)>0, x0 in (a,b), then y(x)>0 for all x in (a,b)

Fibonacci88 said:
Given the equation y'= xg(x,y) , suppose that g and (partial) dg/dy are defined and continuous for all (x,y). Show the following:

1) y(x)=0 is a solution
You can't- it's not true. For example, supose g(x,y)= 1 so the equation is y'= x. Then $y(x)= (1/2)x^2+ C$. y(x)= 0 is not a solution since then y'= 0 and so $y'= 0\ne x$. there may be some other condition that you have left out.

2)if y=y(x), x in (a,b) is a solution and if y(x0)>0, x0 in (a,b), then y(x)>0 for all x in (a,b)

sorry for mistake. I am sending the correct version of the problem now.

i need answer for (ii) and (iii)

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1. What is the "Proof of Solutions for y' = xg(x,y) Equation"?

The "Proof of Solutions for y' = xg(x,y) Equation" is a mathematical process used to verify the existence and uniqueness of solutions to the differential equation y' = xg(x,y). It involves using various techniques such as substitution, separation of variables, and integrating factors.

2. Why is the proof of solutions important?

The proof of solutions is important because it provides a rigorous and systematic approach to solving differential equations. It ensures that the solutions obtained are valid and unique, which is crucial in many scientific and engineering applications.

3. What are the key steps in the proof of solutions?

The key steps in the proof of solutions for y' = xg(x,y) equation include transforming the equation into a separable form, solving for the general solution, and then verifying the initial conditions to find the particular solution. Other techniques such as integrating factors and substitution may also be used depending on the specific equation.

4. Can the proof of solutions be applied to all differential equations?

No, the proof of solutions is specific to differential equations of the form y' = xg(x,y). Other types of differential equations may require different methods of solution.

5. Are there any limitations to the proof of solutions?

The proof of solutions relies on certain assumptions and techniques, and may not be applicable to all scenarios. It is important to carefully consider the initial conditions and any potential singularities in the equation before using the proof of solutions method.

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