[tex]pentazoid said:Homework Statement
Derived the general solution of the given equation by using an appropriate change of variables
5.) du/dt-2(du/dx)=2
Homework Equations
du/dx=du/d(alpha)*d(alpha)/dx+du/d(beta)*d(beta)/dx
du/dt= du/d(alpha)*d(alpha)/dt+du/d(beta)*d(beta)/dt
The Attempt at a Solution
not really sure how to determine du/d(alpha),d(alpha)/dx,d(beta)/dx ,d(alpha)/dt,d(beta)/dt
book says alpha=x+2t, beta=x
the1ceman said:[tex]
\frac{\partial\alpha}{\partial x}=1,\frac{\partial\beta}{\partial x}=1 so
\[
\frac{\partial u}{\partial x}=\frac{\partial u}{\partial\alpha}\frac{\partial\alpha}{\partial x}+\frac{\partial u}{\partial\beta}\frac{\partial\beta}
{\partial x}=\frac{\partialu}{\partialx}=\frac{\partial u}{\partial\alpha}+\frac{\partial u}{\partial\beta}
\]
[/tex]
did that help?
A partial differential equation (PDE) is a mathematical equation that involves partial derivatives of an unknown function with respect to multiple independent variables. It describes the relationship between the function and its derivatives in a particular system.
A partial differential equation involves partial derivatives with respect to multiple independent variables, while an ordinary differential equation involves derivatives with respect to a single independent variable. PDEs are commonly used to model physical phenomena in multiple dimensions, while ODEs are used for single-dimensional problems.
Partial differential equations have a wide range of applications in various fields such as physics, engineering, economics, and biology. They are commonly used to model phenomena such as heat transfer, fluid dynamics, electromagnetism, and quantum mechanics.
There is no general method for solving all types of partial differential equations. The approach to solving a PDE depends on its type and specific form. Some common techniques include separation of variables, method of characteristics, and numerical methods such as finite difference or finite element methods.
Yes, partial differential equations are crucial in modern science and have been instrumental in many scientific discoveries and technological advancements. They are used to model complex systems and make predictions about the behavior of physical phenomena, making them essential tools for scientists and engineers.