Partial diff eqn problem

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In summary: Applying the chain rule, we get \frac{\partial f(x,y)}{\partial x} = \frac{\partial g(s(x), t(y))}{\partial s} \frac{\partial s(x)}{\partial x} now substitute the expressions for s(x) and g(s,t) \frac{\partial f(x,y)}{\partial x} = \frac{\partial g(s(x), t(y))}{\partial s} \frac{1}{x} now similarly take the partial w.r.t y \frac{\partial f(x,y)}{\partial y} = \frac{\partial g(s(x), t(y))}{\partial t} \frac{1}{y} now
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CrimsnDragn
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Homework Statement


the substitution x=es, y=et converts f(x,y) into g(s,t) where g(s,t)=f(es, et). If f is known to satisfy the partial differential equation

x2(d2f/dx2) + y2(d2f/dy2) + x(df/dx) + y(df/dy) = 0

show that g satisfies the partial-differential equation

(d2g/ds2) + (d2g/dt2) = 0

The Attempt at a Solution



I feel like this is a simple problem - All I need to figure out is how to find the partial derivative of f(x,y) and the double partial derivative, but I'm not sure how to do it. Are the values of x and y relevant for the first part of the question?

what would df/dx be? D1f(x,y) = (y)f'(x,y) by the chain rule? and similarly df/dy = D2f(x,y) = (x)f'(x,y)? Someone please help!
 
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  • #2
i assume they're meant to be partials... & I don't really understand what you've written at the end of the post but here's some ideas I hope help

now as you're given
[tex] x(s)=e^s[/tex]
[tex] y(t)=e^t[/tex]

invert these and then re-write as
[tex] s(x)=ln(s)[/tex]
[tex] t(y)=ln(y)[/tex]

so the relation function g & f is given as
[tex] g(s,t) = f(x(s), y(t)) [/tex]

we can re-write it and think of it as
[tex] f(x,y) = g(s(x), t(y)) [/tex]

now try taking a partial w.r.t x and using the chain rule on the RHS

[tex] \frac{\partial f(x,y)}{\pratial x} = \frac{\partial }{\partial x} g(s(x), t(y)) [/tex]
the partial w.r.t. x means the other variable (y) in this case is kept constant
 

1. What is a partial differential equation (PDE)?

A PDE is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe the relationship between a function and its partial derivatives with respect to those variables.

2. What types of problems can be solved using PDEs?

PDEs are commonly used in physics, engineering, and other scientific fields to model and solve problems involving continuous systems, such as heat transfer, fluid dynamics, and quantum mechanics.

3. How do you solve a partial differential equation problem?

The specific method for solving a PDE problem depends on the type of equation and boundary conditions. Common techniques include separation of variables, Fourier series, and numerical methods such as finite differences or finite elements.

4. What are some real-life applications of PDEs?

PDEs have a wide range of applications in many fields, such as modeling weather patterns, predicting financial markets, and designing complex engineering systems like airplanes and bridges.

5. Are there any limitations to using PDEs?

While PDEs are powerful tools for solving many problems, they have limitations. They may not always have analytical solutions, and numerical methods can be computationally intensive. In addition, the assumptions and simplifications made in modeling a problem with PDEs may not accurately reflect real-world situations.

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