Solving a First Order Linear PDE

Click For Summary
SUMMARY

The discussion focuses on solving the first order linear partial differential equation (PDE) given by ∂δ/∂t = (ρg/μ) δ² ∂δ/∂z. The solution derived is δ(z, t) = √(μz)/(ρgt). Participants seek a detailed, step-by-step explanation of how to arrive at this solution. The conversation emphasizes the importance of understanding the transformation of the PDE into its solution form.

PREREQUISITES
  • Understanding of partial differential equations (PDEs)
  • Familiarity with first order linear PDEs
  • Knowledge of mathematical concepts such as separation of variables
  • Basic proficiency in calculus and differential equations
NEXT STEPS
  • Study the method of characteristics for solving first order PDEs
  • Learn about separation of variables in PDEs
  • Explore the derivation of solutions for linear PDEs
  • Investigate applications of PDEs in physics and engineering contexts
USEFUL FOR

Students, mathematicians, and engineers who are working with partial differential equations, particularly those seeking to deepen their understanding of first order linear PDEs and their solutions.

sagigirl
Messages
1
Reaction score
0
Good day. I was wondering if you could help me solve this first order linear partial differential equation:

[∂δ]/[/∂t] = [ρg]/[/μ] δ^2 [∂δ]/[/∂z].

The solution for this is:

δ(z, t) = √[μ z]/[/ρg t].

I don't really understand how the PDE became like this. If you could show the step-by-step solution, I would really, gladly appreciate it. Thank you :)
 
Physics news on Phys.org
sagigirl said:
Good day. I was wondering if you could help me solve this first order linear partial differential equation:

[∂δ]/[/∂t] = [ρg]/[/μ] δ^2 [∂δ]/[/∂z].

The solution for this is:

δ(z, t) = √[μ z]/[/ρg t].

I don't really understand how the PDE became like this. If you could show the step-by-step solution, I would really, gladly appreciate it. Thank you :)

Welcome to the PF.

Is this for schoolwork?
 

Similar threads

Undergrad Maxwell's equations PDE interdependence and solutions
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Stuck on solving a non homogenous (linear) PDE
  • · Replies 36 ·
2
Replies
36
Views
6K
Undergrad Looking for what this type of PDE is generally called
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Classification of First Order Linear Partial Differential Eq
  • · Replies 1 ·
Replies
1
Views
3K
Graduate How to solve simple 2D space-time PDE numerically
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Generic Solution of a Coupled System of 2nd Order PDEs
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad What happens if you divide by a differential?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Heat Equation: Solve with Non-Homogeneous Boundary Conditions
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Spectral Theorem to Convert PDE into ODE
  • · Replies 2 ·
Replies
2
Views
3K