Linear Partial Differential Equation-Problem in solving?

In summary: MitropolskiyIn summary, the conversation discusses a problem with solving a linear partial differential equation using the method of characteristics. The speaker suggests using the operator method as an alternative solution. The equation is rewritten in a different form, and a solution is provided using the operator form. The final solution is given as z(x,y) = F(y-x*(3/2))+x^2/8+xy/2+x/2.
  • #1
shaiqbashir
106
0
Linear Partial Differential Equation---Problem in solving??

Hi guys!

im getting stuck in solving the following Linear Partial Differential Equation:


2p+3q=x+y+1

now when i just form its auxiliary equation it becomes

dx/2 = dy/3 = dz/(x+y+1)

Now the problem that I am just facing is how to solve these three sides. Usually i try to divide and multiply by some constant and then add these three. But in this case, i just couldn't find a way to go!

please help me as soon as possible!

thanks in advance!
 
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  • #2
You are trying to solve linear first order PDE by the method of characteristics and this way leads certainly to the right answer. However there exist many different methods for solving the problem. I would like to point out the uncommon method - operator method.

If rewrite your equation in the following form

dz/dx =- 3/2*dz/dy + f(x,y)

where f(x,y)=(x+y+1)/2

then its solution in the operator form (See http://arxiv.org/abs/math-ph/0409035 ) follows immediately as

z(x,y) = exp(-x*(3/2)*d/dy)*F(y)+int_0^x[exp(-(x-t)*(3/2)*d/dy)* f(t,y)] dt
where F(y) is an arbitrary function.

Since the operator exp(-x*(3/2)*d/dy) here is the simple shift operator we arrive to

z(x,y) = F(y-x*(3/2))+int_0^x[f(t,y-(x-t)*(3/2))] dt

=F(y-x*(3/2))+x^2/8+xy/2+x/2 .


Yurii
 
  • #3


There are a few potential issues that could be causing difficulty in solving this linear partial differential equation. First, it's important to note that the auxiliary equation you have formed is not correct. The correct auxiliary equation for this equation would be dx/2 = dy/3 = dz/1. This is because the coefficients of x and y are both 1, so they should be represented as 1 in the auxiliary equation.

Another potential issue is that the equation you have given is not a partial differential equation. It is a system of linear equations with two unknowns, x and y. This means that there is no need to form an auxiliary equation or use any special methods for solving partial differential equations. Instead, you can simply solve for x and y using standard algebraic techniques, such as substitution or elimination.

If you are trying to solve a different linear partial differential equation and are having trouble, it could be due to the complexity of the equation itself. Linear partial differential equations can be quite challenging to solve, especially if they involve higher order derivatives or non-constant coefficients. In these cases, it may be helpful to use specialized techniques, such as separation of variables or the method of characteristics, or to seek assistance from a tutor or classmate. Good luck with your studies!
 

1. What is a linear partial differential equation?

A linear partial differential equation is a mathematical equation that involves partial derivatives of a dependent variable with respect to multiple independent variables, and has a linear relationship between the dependent variable and its derivatives. It is commonly used to model physical phenomena in physics, engineering, and other fields.

2. What are the common difficulties in solving linear partial differential equations?

Some common difficulties in solving linear partial differential equations include finding the appropriate boundary and initial conditions, determining the correct method of solution, and dealing with complicated or nonlinear physical systems.

3. What are some techniques for solving linear partial differential equations?

Some common techniques for solving linear partial differential equations include separation of variables, the method of characteristics, and the Laplace transform method. Other techniques such as the finite difference method and the finite element method are also commonly used.

4. How can one check the accuracy of a solution to a linear partial differential equation?

One way to check the accuracy of a solution is by substituting the solution into the original differential equation and verifying that it satisfies the equation. Another way is to compare the solution to known analytical or numerical solutions, if available.

5. What are some real-world applications of linear partial differential equations?

Linear partial differential equations have numerous real-world applications, including modeling heat transfer, fluid dynamics, electromagnetic fields, and quantum mechanics. They are also used in image and signal processing, financial mathematics, and many other fields.

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