Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Calculus and Beyond Homework Help
Applying a substitution to a PDE
Reply to thread
Message
[QUOTE="etotheipi, post: 6324346"] [B]Homework Statement:[/B] Please see below [B]Relevant Equations:[/B] N/A [B]Problem: [/B]Consider the equation $$\frac{\partial v}{\partial t} = \frac{\partial^{2} v}{\partial x^2} + \frac{2v}{t+1}$$ where ##v(x,t)## is defined on ##0 \leq x \leq \pi## and is subject to the boundary conditions ##v(0,t) = 0##, ##v(\pi, t) = f(t)##, ##v(x,0) = h(x)## for some functions ##f(t)## and ##h(x)##. Using the substitution ##v=(t+1)^{2}u##, show that ##u## satisfies $$\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^2}$$ [B]Attempt: [/B]I'm not sure if I'm doing the differentiation correctly. I did $$\frac{\partial v}{\partial t} = 2u(t+1)$$ $$\frac{\partial^{2} v}{\partial x^{2}} = \frac{\partial}{\partial x} \frac{\partial v}{\partial u} \frac{\partial u}{\partial x} = \frac{\partial}{\partial x} (t+1)^{2} \frac{\partial u}{\partial x} = (t+1)^{2} \frac{\partial^2 u}{\partial x ^2}$$ Plugging this in doesn't appear to give the result. My suspicion is that I was supposed to use the product rule for the first derivative, however I don't think that is right since I thought we were supposed to hold everything else constant during the differentiation? If I try this for the sake of it, I get $$\frac{\partial v}{\partial t} = 2u(t+1) + (t+1)^{2} \frac{\partial u}{\partial t}$$ $$\frac{\partial^{2} v}{\partial x^{2}} = \frac{\partial}{\partial x} \frac{\partial v}{\partial u} \frac{\partial u}{\partial x} = \frac{\partial}{\partial x} [(t+1)^{2} \frac{\partial u}{\partial x} + 2u(t+1)][\frac{\partial u}{\partial x}]$$ This seems even more wrong. So I wondered whether anyone could give me a pointer? Thanks! [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Calculus and Beyond Homework Help
Applying a substitution to a PDE
Back
Top