Q2/ Solve differential equation.Uxy = cos(x + y)²

Q2: Given PDE is the following :integrating partially with respect to x,we get.Let,So we have again…

Answered: Q2/Solve differential equation. Uxy =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Q2/ Solve differential equation.
Uxy = cos(x + y)²

Expert Solution

Step 1: PDE solution

Q2: Given PDE is the following :

$U subscript x y end subscript equals cos left parenthesis x plus y right parenthesis squared$

integrating partially with respect to x,we get

$U subscript y equals integral cos left parenthesis x plus y right parenthesis squared d x plus phi left parenthesis y right parenthesis comma w h e r e comma phi space i s space a r b i t r a r y space f u n c t i o n space o f space y space o n l y$ .

Let, $V left parenthesis x comma y right parenthesis equals integral cos left parenthesis x plus y right parenthesis squared d x$

So we have $U subscript y equals V left parenthesis x comma y right parenthesis plus phi left parenthesis y right parenthesis$

again integrating to above with respect to y,we get

$U left parenthesis x comma y right parenthesis equals integral V left parenthesis x comma y right parenthesis d y plus integral phi left parenthesis y right parenthesis d y plus psi left parenthesis x right parenthesis comma w h e r e space psi space i s space a r b i t r a r y space f u n c t i o n space o f space x space o n l y$