Differential Equation Exact Solution

• jesuslovesu
In summary, the given equation can be rewritten as (x-y)^2 = 7-xy, which is equivalent to the quadratic equation y^2 + (-x)y + (x^2-7) = 0. The discriminant of this quadratic equation is 28-3x^2, which leads to the solution y = x + sqrt(28-3x^2)/2.
jesuslovesu

Homework Statement

(2x - y) + (2y-x) dy/dx = 0
y(1) = 3
Solve and determine where the solution is approximately valid.

The Attempt at a Solution

Ux = 2x - y
Uy = 2y - x

U = x^2 - yx + h(y)
Uy = 2y -x = -x + h'(y)
h'(y) = 2y
h(y) = y^2

x^2 - yx + y^2 = c
1 - 3 + 3^2 = c = 7

I'm not really sure where to go from here,
x^2 - yx + y^2 = 7
However, the answer is quite a bit different:
y = x + sqrt( 28 - 3x^2)/2 and I can't quite see the step toward that. (x-y)^2 = 7-xy but I still can't seem to solve for y

Last edited:
I have only read the last line of your post, so I don't know if your steps are right, but regarding solving for y in (x-y)^2 = 7-xy, notice that solving this equation for y(x) is the same as finding the roots of the quadratic equation y^2 + (-x)y + (x^2-7) = 0.

And, in this case "b2- 4ac"= (-x)2- 4(1)(x[sup[2]- 7)= x2- 4x2+ 28= 28- 3x2. That's where that came from.

What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model a wide range of phenomena in physics, engineering, economics, and other fields.

What is an exact solution to a differential equation?

An exact solution to a differential equation is a solution that satisfies the equation for all values of the independent variable. It is obtained by manipulating the equation algebraically, rather than using numerical methods.

How is an exact solution different from an approximate solution?

An exact solution is a solution that satisfies the differential equation exactly, while an approximate solution is an estimated solution that may not be entirely accurate but is close enough for practical purposes. Approximate solutions are often obtained using numerical methods.

What is the process for finding an exact solution to a differential equation?

The process for finding an exact solution varies depending on the type of differential equation. In general, it involves identifying the type of differential equation, manipulating it algebraically to simplify it, and then solving for the dependent variable in terms of the independent variable.

Are there any special techniques for solving differential equations with exact solutions?

Yes, there are several special techniques for solving differential equations with exact solutions, such as separation of variables, integrating factors, and substitution. These techniques can be used to simplify the equation and make it easier to find an exact solution.

• Calculus and Beyond Homework Help
Replies
6
Views
983
• Calculus and Beyond Homework Help
Replies
2
Views
681
• Calculus and Beyond Homework Help
Replies
18
Views
2K
• Calculus and Beyond Homework Help
Replies
7
Views
661
• Calculus and Beyond Homework Help
Replies
1
Views
569
• Calculus and Beyond Homework Help
Replies
10
Views
741
• Calculus and Beyond Homework Help
Replies
25
Views
676
• Calculus and Beyond Homework Help
Replies
2
Views
572
• Calculus and Beyond Homework Help
Replies
3
Views
366
• Calculus and Beyond Homework Help
Replies
24
Views
2K