Partial Differential Equation with square roots

In summary, the conversation discusses a partial differential equation with square roots that the student is struggling to solve. After attempting to solve the equation using the separation of variables method, they are unable to eliminate the square root. The moderator suggests that the solution to their problem is also a solution to a different problem, and clarifies that the separation of variables method can be used to simplify the equation.
  • #1
<Moderator's note: Moved from a technical forum and thus no template.>

Hi everyone,
I have encountered a partial differential equation with square roots which I don't have a clue in solving it. After letting z=F(x)+G(y), I can't really figure out the next step. I tried squaring both sides but the square root still exists. Any help would be appreciated, thank you!

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  • #2
Hello Johnson, :welcome:

Please post in a homework forum and use the template. See Guidelines . We need an attempt at solution (posted) to be allowed to help out.

In the mean time:
How would you solve ##\sqrt{\partial z\over \partial x} = x ## ?
 
  • #3
Thank you for your reply, my attempt to your problem are as follow:
\begin{aligned}\dfrac {\partial z}{\partial x}=x^{2}\\ \partial z=x^{2}\partial x\\ z=\dfrac {x^{3}}{3}+c\end{aligned}
My attempt to my problem are as follow:
I try to reproduce the equation to a 1st order differential equation,
Let z(x,y)= F(x)+G(y)
Therefore the equation becomes (F'(x))^1/2+(G'(y))^1/2=x
Taking square on both sides still doesn't eliminate the the square root and squaring it one more time involves F'(x)^2.
 
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  • #4
Johnson Chou said:
my attempt to your problem
Good. Notice anything remarkable in relation to your problem ?
 
  • #5
Sorry I don't quite understand the relation, but my attempt on continuing the question with your hint:
\begin{aligned}\sqrt {\dfrac {\partial z}{\partial x}}+\sqrt {\dfrac {\partial z}{\partial y}}=x\\
\sqrt {\dfrac {\partial z}{\partial x}}=x\ & \sqrt {\dfrac {\partial z}{\partial y}}=0\\
\therefore \dfrac {\partial z}{\partial x}=x^{2},\dfrac {\partial z}{\partial y}=0\\
z=\dfrac {x^{3}}{3}+G\left( y\right) =\dfrac {x^{3}}{3}+C\end{aligned}
 
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  • #6
That's what I meant: the solution to my problem is also a solution for your problem :smile:
I must say that the separation of variables thing is useful but a bit confusing: you were supposed to conclude that if ##z = F(x) + G(y)## then ##\sqrt{\partial z\over \partial y} ## is a function of ##y## and the right hand side of the equation is only a function of x, square root or no square root on the left. In other words, satisfied if ##G(y) = 0##.
 
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Likes Johnson Chou
  • #7
BvU said:
That's what I meant: the solution to my problem is also a solution for your problem :smile:
I must say that the separation of variables thing is useful but a bit confusing: you were supposed to conclude that if ##z = F(x) + G(y)## then ##\sqrt{\partial z\over \partial y} ## is a function of ##y## and the right hand side of the equation is only a function of x, square root or no square root on the left. In other words, satisfied if ##G(y) = 0##.
Thanks! Helped me out a lot.
 

1. What is a "Partial Differential Equation with square roots"?

A Partial Differential Equation with square roots is a type of mathematical equation that involves two or more independent variables and their partial derivatives, with at least one square root term in the equation. These equations are commonly used in physics and engineering to model complex systems and phenomena.

2. How do you solve a Partial Differential Equation with square roots?

Solving a Partial Differential Equation with square roots requires advanced mathematical techniques and may not have a closed-form analytical solution. However, numerical methods such as finite difference, finite element, and spectral methods can be used to approximate the solution.

3. What are the applications of Partial Differential Equations with square roots?

Partial Differential Equations with square roots have numerous applications in various fields such as fluid dynamics, heat transfer, quantum mechanics, and image processing. They are also used to model complex phenomena such as diffusion, wave propagation, and reaction-diffusion processes.

4. How does a Partial Differential Equation with square roots differ from an ordinary differential equation?

Unlike ordinary differential equations, which involve only one independent variable, Partial Differential Equations with square roots involve multiple independent variables. They also involve partial derivatives, which account for the rate of change in one variable while holding the other variables constant.

5. What are the challenges in solving Partial Differential Equations with square roots?

Partial Differential Equations with square roots can be challenging to solve as they often have non-linear terms and may not have an analytical solution. Additionally, the use of multiple independent variables and partial derivatives increases the complexity of the problem and requires advanced mathematical techniques for solving.

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