Hi,I need help in PDE homework. There are 10 problems. Page 71, Problem 4.1 (a, b, c) For 4.1-c,...

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Hi,I need help in PDE homework. There are 10 problems. Page 71, Problem 4.1 (a, b, c)  For 4.1-c, What happens if you try to solve it (Choose one method and try to solve it).  Page 75, Problems 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, (5.8 for 5.6 only) Please Read the lecture notes to know how to solve the problem like the teacher style.I need step by step answer for integral and differential.Also check your solution for u1 and u2Also grad(u1)Xgrad(u2)And the check zTake a look at the attached file.

Robert · Accepted Answer

(a) For the given equation and the initial curve C: x=t, y=t, t>0 we can write the ordinary differential 
equation can be written as,
Given that z = 2t on Curve C , so the above equation will become 
1/z = 1/y = 2/x = μ 
So from theorem 4.2 we have infinitely many solutions 
So, x = 2 μ , y = μ, z = μ, since μ is a constant number then it can have multiple values so infinite 
solutions exits.
(b) For z = 1 on C
Given that z = t  
1/z = 1/y = 1/x = μ 
So from theorem 4.2 we have infinitely many solutions 
So, x = μ, y = μ, z = μ , since μ is a constant number then it can have multiple values so infinite solutions 
exits.
(c) z = sin(pi*t/2 ) on Curve C
Given that z = sin(pi*t/2 ) 
1/z = 1/y = cos(pi*t/2 )/x = μ 
1/z = 1/y = 0/x = μ 
So, here μ = 0  
So from theorem 4.2 we have unique solution. 
So, x = 0, y = 0 , z = 0  
Hence unique solution exits.
Answer 
Given partial differential equation is,
Where f and a are constant.  
If we need to find a solution to the problem we can consider the case of ordinary differential equation and 
the associated equation can be written as,
To solve this system of equation we need to take two independent functions u1 and u2 as, 
u1 = z and u2 = x –a(z)y 
So on solving the above differential equation for these two systems we get, 
z = F(x-a(z)y) 
If we apply the boundary condition then we get, 
F(x) = f(x) 
 Now if we use the implicit function theorem for the given equation we see that at point(x0,y0) and the 
boundary condition z(x,

Hi, I need help in PDE homework. There are 10 problems. Page 71, Problem 4.1 (a, b, c) For 4.1-c, What happens if you try to solve it (Choose one method and try to solve it). Page 75, Problems 5.1,...

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