Solution of the 2nd-order pde u_t=u_xy

Thread starter pep2010
Start date Jan 13, 2010
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Pde

In summary, the 2nd-order partial differential equation u_t = u_xy can be solved using the method of characteristics. This involves rewriting the equation in the form of a system of first-order equations and solving for the characteristic curves using the initial conditions. The constants of integration in the solution represent arbitrary functions that can be determined using the initial conditions, and the solution can be written in a closed form depending on the initial conditions and the specific form of the equation. This type of equation has many applications in fields such as physics, engineering, and economics, including modeling diffusion processes, heat conduction, and population dynamics.

Jan 13, 2010

pep2010

hey guys,

i've reduced a more complex pde to the second-order linear equation u_t=u_xy, but now I'm a bit stuck!

firstly, does anyone know if this equation has a proper name and thus been studied somewhere in the literature?

secondly, any ideas on how to proceed with the general solution?

cheers, pep2010

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Jan 13, 2010

gato_

This a rotated version of the heat equation. Check for that one

1. How do you solve the 2nd-order partial differential equation u_t = u_xy?

The solution to this equation involves using the method of characteristics. First, we rewrite the equation in the form of a system of first-order equations by introducing new variables s and t, such that s = x + y and t = x - y. Then, we solve for the characteristic curves using the initial conditions. Finally, we plug in the characteristic curves into the original equation to get the solution u(x,y).

2. What are the initial conditions required for solving this equation?

The initial conditions needed for solving this equation are the initial values of u(x,y) and its partial derivatives with respect to x and y at a given point (x0,y0). These initial conditions will determine the characteristic curves and help us find the solution u(x,y).

3. What is the significance of the constants of integration in the solution?

The constants of integration represent the arbitrary functions that arise during the process of solving the equation using the method of characteristics. These functions can be determined using the initial conditions, and they help us get a unique solution for u(x,y).

4. Can the solution of u_t = u_xy be written in a closed form?

Yes, the solution of this equation can be written in a closed form using the method of characteristics. However, the form of the solution will depend on the initial conditions and the specific form of the equation.

5. Are there any applications of this type of partial differential equation in real-world problems?

Yes, this type of partial differential equation has many applications in various fields such as physics, engineering, and economics. It can be used to model diffusion processes, heat conduction, and population dynamics, among others.