Solwant to solve a 2nd ving a partial differential equation with textbook solution

09152023, 08:56 AM
Post: #1




Solwant to solve a 2nd ving a partial differential equation with textbook solution
First I managed to solve a 2nd order ode using the symbolic letters as in:
desolve(∂(∂u/∂x)∂x+3*∂u/∂x=0,x,y) Now want to solve a 2nd order partial differential equation, pde My Reference for a textbook solution is from the book "Advanced Mathematics For Engineering Students", page 131. In the textbook The 2nd order pde is ∂2u/(∂x∂y) = 2x  y. The general solution is: u = x^2*y  1/2*x*y^2 + F(x) + G(y) Where: F(x) = 2* sin(x) and G(y) = 3*y^4 5 Entering the pde in the Prime [/b][/align]n CAS mode: b]desolve(∂(∂u/∂y)∂x+y2*x=0,x,y,u)[/b] The answer is [1/2*y] It was not the textbook's answer. Question: why didn't the output from the prime match the textbook answer?[b] Thank you, Anthony, Sydney 

09152023, 12:18 PM
Post: #2




RE: partial differential equation with textbook solution
(09152023 08:56 AM)Anthony The Koala Wrote: In the textbook F(x) and G(y) can be any expression, that particular set is just for illustration. diff(u, y, x), both F(x) and G(y) goes away, leaving just (2x  y) From XCas manual, desolve(de, <x, y>), where x is variable, y is unknown. desolve may not be able to handle more variables. (it has other limits, see manual) We do this in steps (1 variable, 1 unknown). let w = diff(u,y) Cas> desolve(diff(w, x) = (2*x  y), x, w) x*y + x^2 + G_0 Cas> desolve(diff(u, y) = (x*y + x^2), y, u) Cas> expand(Ans) 1/2*x*y^2 + x^2*y + G_0 Or, we simplify integrate twice, to get u Cas> int(int((2*x  y), x), y) 1/2*x*y^2 + x^2*y u = x^2*y  1/2*x*y^2 + F(x) + G(y), same as textbook answer 

09162023, 01:56 AM
Post: #3




RE: Solwant to solve a 2nd ving a partial differential equation with textbook solution
Dear Mr Chan,
Thank you for the reply. It is a learning experience of the limitations of the Prime not being able to handle PDEs. I tried your twostage solution and it works. I wanted to go deeper by breaking down the statement desolve(diff(w, x) = (2*x  y), x, w) and do diff(w, x) = (2*x  y), x, w) The display said [diff(w,x)=2*xy x w] [0=2*xy x w] I tried b]diff(w, x) = (2*x  y))[/b] The display said [diff(w,x)=2*xy] [0=2*xy] I know desolve solves an ode but need further idea of what the Ans within the desolve does with [diff(w,x)=2*xy x w] [0=2*xy x w] ? [/b] Ref for the int command http://www.wiki4hp.com/doku.php?id=prime...s_commands Other question please Obviously there are many ways to solve a PDE including taking Laplace transforms and converting them to an ODE. Could the twostep method used in the original question be used to solve any kind of PDE and the need to specify the initial and boundary conditions? Thank you Anthony, Sydney[/i] 

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