Solve this PDE 2Ux-Uy+5U=10 with U(x,0)=0

[BustedBreaks]

I'm following an algorithm my teacher gave us and I'm trying to understand it...

I'm trying to solve this PDE 2Ux-Uy+5U=10 with U(x,0)=0

First I need to solve the homogeneous equation.

So I set up the relation:
V(y)=U(2y+c, y)

to solve 2Ux-Uy=0

where the characteristic equation is y=1x/2 -c/2 and x=2y+c

***My first problem is that I am having a hard time understanding what the characteristic equation is and why it is important to use in solving this problem.

Then I need to to take the derivative of V(y) with respect to y

V ' (y) =(Ux)(Xy) + (Uy)(Yy) which is
V ' (y) =2Ux+Uy

here is my second problem, I seem to get 2Ux+Uy instead of 2Ux-2Uy which is what I need to make the problem a little easier...EDIT... Finally got it, nvm...

 

[mighty2000]

I understand that you are trying to solve the PDE 2Ux-Uy+5U=10 with the initial condition U(x,0)=0. I will try to explain the concept of characteristic equation and why it is important in solving this problem.

In solving a PDE, we often use the method of characteristics to find a solution. The characteristic equation is a set of equations that describes the behavior of a PDE along a particular curve or surface. In other words, it helps us to find a solution by identifying a family of curves or surfaces that satisfy the PDE.

In your case, the characteristic equation is y=1x/2 -c/2 and x=2y+c. This equation is obtained by setting the coefficient of the highest order derivative (in this case, Ux) equal to 0. Solving this equation will give us a family of curves that satisfy the PDE.

Now, when you take the derivative of V(y) with respect to y, you will get 2Ux+Uy. This is because the characteristic equation involves both x and y variables, so when you take the derivative with respect to y, x is treated as a constant. Therefore, you will get 2Ux+Uy instead of 2Ux-2Uy.

I hope this explanation helps you understand the concept of characteristic equation and why it is important in solving this PDE. Keep up the good work in trying to understand the algorithm given by your teacher. If you have any further questions, feel free to ask. Keep learning and exploring!