Differential Equation Solution
Date: Fall 2013
I have been trying to solve the 1st order partial differential equation 4 (dz/dx) + 5 (dz/dy)=10y with the initial condition z(x,0)=tan x.
I can arrive at the general solution with ease, but the particular solution presents a problem.
Here is the general solution: z-5y^2 =f((1/4)x-y) or you can rewrite this as z= f((1/4)x-y)+5y^2 . However, once I try to substitute in my initial condition, I end up with f((1/4)x)=tan x . I do not know what to do from here. Can someone please help me?
P.S. This is not a homework question. I am an 11th grader in AP calculus AB so this stuff will not be taught in my class, but I have taught myself how to solve various ordinary differential equations. I am now trying to teach myself partial differential equations.
Thanks for the question. What you have provided is not complete. Could you please define "f" used below? Is z = f(x,y), as is used typically? I would like to see a scan of your work as it will better help me to answer your question. If you get f((1/4x) - y) = tan(x), you may have a transcendental equation which cannot be solved in closed form. You can solve a transcendental equation numerically. I would encourage you to look up transcendental equations in your AP calculus book as you will encounter them frequently in physics and engineering.
I hope this helps. Please let me know if you have more questions.
Click here to return to the Mathematics Archives
Update: November 2011