- #1
pentazoid
- 146
- 0
Homework Statement
which solutions of du/dt+du/dx=0 is equal to xe-x2
Homework Equations
The Attempt at a Solution
u(x,0) = xe-x2
u(x,t)= (x-t)e(-x-t)2
what else do i need to do?
One way of solving a pde like that is to always try separation of variables , so that you assume[tex]u(x,t)=X(x)T(t)[/tex]where X is just a function of x and T just a function of t, plug this into your pde and see what u get!pentazoid said:Homework Statement
which solutions of du/dt+du/dx=0 is equal to xe-x2
Homework Equations
The Attempt at a Solution
u(x,0) = xe-x2
u(x,t)= (x-t)e(-x-t)2
what else do i need to do?
A PDE, or partial differential equation, is a type of mathematical equation that involves multiple independent variables and their partial derivatives. It is commonly used to describe physical phenomena in fields such as physics, engineering, and finance.
To solve a PDE problem, you need to first identify the type of PDE (e.g. elliptic, parabolic, hyperbolic) and then choose an appropriate solution method. In this case, the given PDE is a first-order linear parabolic equation, which can be solved using the method of characteristics.
The initial condition, u(x,0)=xe-x2, represents the value of the unknown function u at time t=0. It is a boundary condition that helps to determine the unique solution to the PDE problem.
The method of characteristics involves finding characteristic lines, which are curves along which the PDE reduces to an ordinary differential equation (ODE). In this problem, the characteristic lines are given by dx/dt=1 and du/dt=-1. By solving these ODEs, we can obtain the solution u(x,t)=f(x-t)e-t, where f is an arbitrary function.
The solution u(x,t)=f(x-t)e-t represents the evolution of the unknown function u(x,t) over time, starting from the given initial condition. It can be used to study the behavior of a physical system described by the PDE and make predictions about its future state.