The discussion focuses on solving the partial differential equation (PDE) given by du/dt - 2(du/dx) = 2 using a change of variables. The variables alpha and beta are defined as alpha = x + 2t and beta = x, which are crucial for transforming the PDE. Participants express confusion regarding the determination of the derivatives du/d(alpha), d(alpha)/dx, d(beta)/dx, d(alpha)/dt, and d(beta)/dt. The discussion emphasizes the importance of understanding the relationships between these variables to derive the general solution effectively.
PREREQUISITESStudents studying mathematics, particularly those focusing on differential equations, as well as educators and tutors seeking to clarify concepts related to PDEs and variable transformations.
<br /> \frac{\partial\alpha}{\partial x}=1,\frac{\partial\beta}{\partial x}=1 so<br /> \[<br /> \frac{\partial u}{\partial x}=\frac{\partial u}{\partial\alpha}\frac{\partial\alpha}{\partial x}+\frac{\partial u}{\partial\beta}\frac{\partial\beta}<br /> {\partial x}=\frac{\partialu}{\partialx}=\frac{\partial u}{\partial\alpha}+\frac{\partial u}{\partial\beta}<br /> \]<br />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:<br /> \frac{\partial\alpha}{\partial x}=1,\frac{\partial\beta}{\partial x}=1 so<br /> \[<br /> \frac{\partial u}{\partial x}=\frac{\partial u}{\partial\alpha}\frac{\partial\alpha}{\partial x}+\frac{\partial u}{\partial\beta}\frac{\partial\beta}<br /> {\partial x}=\frac{\partialu}{\partialx}=\frac{\partial u}{\partial\alpha}+\frac{\partial u}{\partial\beta}<br /> \]<br />
did that help?