How can the 2nd Order Elliptic Equation be solved for exact solutions?

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In summary, the conversation discusses the search for an exact solution to the equation d2y/dx2 = C/y, where C is a constant. The equation arises when deriving the equation of motion in a gravitational field. The individual is interested in solving the equation and has only managed to express it as a power series using Taylor's theorem. They inquire about any special functions or approximations that can solve the equation. Another individual suggests a trick for solving second-order equations of this type and provides an example of how to apply it to this particular equation. The individual expresses their appreciation for the quick replies and acknowledges the difficulty of solving the equation.
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Noctisdark
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I've been searching for exact solution of d2y/dx2 = C/y, where C is some constant, such equation take place when deriving equation of motion in gravitationnal field, I'm more interested in how to solve it, yet I only managed to express it as power series using taylor's theorem at x = 0, just pick y(0) = c1, y'(0) = c2, so y''= C/c1, y'''(x) = C*c2/c12 and so on, until y ≈ c1 + c2*x + C/2c1 * x2+ C*c2/6c12 *x3 + ..., Is there's any special function that solves the equation or any other approximisation ?, I want to hear some thoughts !,
 
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  • #2
If we let ##v = dy/dx##, then

$$\frac{C}{y} = \frac{d^2y}{dx^2} = \frac{dv}{dx} = \frac{dy}{dx} \frac{dv}{dy} = v \frac{dv}{dy}.$$

We can integrate this for ##v(y)## (it's the square root of log). Solving ##v = dy/dx## for ##y## in closed form is probably not possible.
 
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There is a trick for solving second order equations of this type, where the independent variable x is missing. Using the notation y' = dy/dx, write d2y/dx2 as follows:
[tex] \frac{d^2 y}{dx^2} = \frac{dy'}{dx} = \frac{dy'}{dy} \frac{dy}{dx} = y' \frac{dy'}{dy}[/tex]

Now you have eliminated x from the equation and turned it into a first order equation in y and y'. For your equation:

[tex] y' \frac{dy'}{dy} = \frac{C}{y} [/tex]

or:

[tex] y' dy'= \frac{C}{y} dy[/tex]

Now you can integrate both sides, and solve for y' in terms of y. Then you replace y' with dy/dx and integrate a second time. So you should be able to solve your equation analytically without needing power series.
 
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Thanks for quick replies, as far as I tried this seems "Damn" hard to solve but can get other approximisation from it , thanks :p
 

1. What is a 2nd Order Elliptic Equation?

A 2nd Order Elliptic Equation is a type of partial differential equation that involves a second-order derivative and has the general form: auxx + buxy + cuyy + dux + euy + fu = g, where a, b, c, d, e, and f are coefficients and g is a function of x and y.

2. What is the meaning of "elliptic" in 2nd Order Elliptic Equation?

The term "elliptic" refers to the shape of the curve that is formed when the equation is graphed. In contrast to parabolic or hyperbolic equations, elliptic equations produce curves that are closed and do not extend to infinity. This is due to the fact that the coefficients a, b, and c all have the same sign.

3. What are some real-world applications of 2nd Order Elliptic Equations?

2nd Order Elliptic Equations have a wide range of applications in fields such as physics, engineering, and finance. They are commonly used to model heat transfer, diffusion, and fluid flow. They can also be used to solve optimization problems, as well as in the pricing of financial derivatives.

4. How is a 2nd Order Elliptic Equation solved?

There are several methods for solving 2nd Order Elliptic Equations, including the finite difference method, the finite element method, and the boundary element method. These methods involve discretizing the equation and solving it numerically. Alternatively, certain types of 2nd Order Elliptic Equations have analytical solutions that can be obtained using techniques such as separation of variables or Laplace transforms.

5. What are the boundary conditions for a 2nd Order Elliptic Equation?

The boundary conditions for a 2nd Order Elliptic Equation specify the values of the dependent variable u at the boundaries of the domain. These conditions are necessary for obtaining a unique solution to the equation. Common types of boundary conditions include Dirichlet, Neumann, and Robin boundary conditions, which involve specifying the value of u, the value of its derivative, or a combination of both at the boundary points.

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