1. ## Differentiation

In my assignment, I got a few questions similar to the one shown below. I know I have to use partial differentiation but I have no idea where to start given these particular kind of questions.If you show me how to do these two, I think I'll do the rest at ease. Your help will be highly appreciated.

1.Determine the values of a and b in the functions
u(x,y) = 16x^3 - 48xy^2+16x^2 - 16y^2, v(x,y) = ax^2y -16y^3 +b xy such that u and v satisfy the Cauchy-Riemann equations
u/ x
= v/ y
u/ y
= - v/ x
.
2.Determine the POSITIVE values of a and b in the function
u(x,t) = e^(x+ at)(6x+bt) such that u satisfies the wave equation
^2 u/ t^2
= 17^2 (^2 u/ x^2)
.

2. Originally Posted by McCoy
In my assignment, I got a few questions similar to the one shown below. I know I have to use partial differentiation but I have no idea where to start given these particular kind of questions.If you show me how to do these two, I think I'll do the rest at ease. Your help will be highly appreciated.

1.Determine the values of a and b in the functions
u(x,y) = 16x^3 - 48xy^2+16x^2 - 16y^2, v(x,y) = ax^2y -16y^3 +b xy such that u and v satisfy the Cauchy-Riemann equations
¶/u x
= v y
u y
= - v x
.
2.Determine the POSITIVE values of a and b in the function
u(x,t) = ex+ at(6x+bt) such that u satisfies the wave equation
^2 u t^2
= 172 2 u x^2
.
Do the partial derivatives, plug into the equations and then the coefficients of like powers of x and y (x and t in the second question) must be equal. That will give you a set of simultaneous equations for the unknowns independent of x and y which you should solve.

In fact for question 1, you will obtain a solution for a and b from the first of the C-R equations, and you will just need to show that this is also a solution for the second

RonL

3. Originally Posted by McCoy
In my assignment, I got a few questions similar to the one shown below. I know I have to use partial differentiation but I have no idea where to start given these particular kind of questions.If you show me how to do these two, I think I'll do the rest at ease. Your help will be highly appreciated.

2.Determine the POSITIVE values of a and b in the function
u(x,t) = ex+ at(6x+bt) such that u satisfies the wave equation
2 u t2
= 172 2 u x2
.
I take it that you mean to find a and b such that $u(x,t)=ex+at(6x+bt)$ is a solution to $\frac{\partial^2u}{\partial t^2}=172\frac{\partial^2u}{\partial x^2}$?

well, find $\frac{\partial^2u}{\partial t^2}$ and $\frac{\partial^2u}{\partial t^2}$.

$u(x,t)=ex+6axt+abt^2$

$\frac{\partial u}{\partial t}=6ax+2abt$

$\frac{\partial^2 u}{\partial t^2}=2ab$

$\frac{\partial u}{\partial x}=e+6at$

$\frac{\partial^2 u}{\partial x^2}=0$

Thus, $2ab=0$...

...are you sure you wrote out the problem right?

I'm getting zero as an answer for both!

--Chris

4. Originally Posted by Chris L T521
I take it that you mean to find a and b such that $u(x,t)=ex+at(6x+bt)$ is a solution to $\frac{\partial^2u}{\partial t^2}=172\frac{\partial^2u}{\partial x^2}$?

well, find $\frac{\partial^2u}{\partial t^2}$ and $\frac{\partial^2u}{\partial t^2}$.

$u(x,t)=ex+6axt+abt^2$

$\frac{\partial u}{\partial t}=6ax+2abt$

$\frac{\partial^2 u}{\partial t^2}=2ab$

$\frac{\partial u}{\partial x}=e+6at$

$\frac{\partial^2 u}{\partial x^2}=0$

Thus, $2ab=0$...

...are you sure you wrote out the problem right?

I'm getting zero as an answer for both!

--Chris

May-be $ex$ means $e^x$?

Actually I see that the original poster has modified the question.

RonL

5. Originally Posted by Chris L T521
I take it that you mean to find a and b such that $u(x,t)=ex+at(6x+bt)$ is a solution to $\frac{\partial^2u}{\partial t^2}=172\frac{\partial^2u}{\partial x^2}$?

well, find $\frac{\partial^2u}{\partial t^2}$ and $\frac{\partial^2u}{\partial t^2}$.

$u(x,t)=ex+6axt+abt^2$

$\frac{\partial u}{\partial t}=6ax+2abt$

$\frac{\partial^2 u}{\partial t^2}=2ab$

$\frac{\partial u}{\partial x}=e+6at$

$\frac{\partial^2 u}{\partial x^2}=0$

Thus, $2ab=0$...

...are you sure you wrote out the problem right?

I'm getting zero as an answer for both!

--Chris
Thanks guys but I'm sorry the way I've typed the questions was wrong. I have done some corrections just now.e.g for the second question, it's 17^2, not 172 .e^(x+at), not ex. I'm sorry guys for the inconvenience.

6. Originally Posted by McCoy
In my assignment, I got a few questions similar to the one shown below. I know I have to use partial differentiation but I have no idea where to start given these particular kind of questions.If you show me how to do these two, I think I'll do the rest at ease. Your help will be highly appreciated.

2.Determine the POSITIVE values of a and b in the function
u(x,t) = e^(x+ at)(6x+bt) such that u satisfies the wave equation
^2 u/ t^2
= 17^2 (2 u/ x^2)
.
I figured you were missing some sort of exponential function here.

Well, you still need to find $\frac{\partial^2u}{\partial t^2}$ and $\frac{\partial^2u}{\partial x^2}$

$\because u(x,t)=e^{x+at}(6x+bt)$

$\frac{\partial u}{\partial t}=e^{x+at}(b)+ae^{x+at}(6x+bt)=(b+6ax+abt)e^{x+at }$

$\frac{\partial^2u}{\partial t^2}=e^{x+at}(ab)+ae^{x+at}(b+6ax+abt)=(2ab+6a^2x+ a^2bt)e^{x+at}$

$\frac{\partial u}{\partial x}=e^{x+at}(6)+e^{x+at}(6x+bt)=(6+6x+bt)e^{x+at}$

$\frac{\partial^2 u}{\partial x^2}=e^{x+at}(6)+e^{x+at}(6+6x+bt)=(12+6x+bt)e^{x+ at}$

Thus,

$(2ab+6a^2x+a^2bt)e^{x+at}=17^2(12+6x+bt)e^{x+at}$

$\implies 2ab+6a^2x+a^2bt=3468+1734x+17^2bt)$

We can come up with 3 simultaneous equations equations:

$\left\{\begin{array}{lcl}
2ab & = & 3468\\
6a^2 & = & 1734 \\
a^2b & = & 17^2b \\
\end{array}
\right.
$

It turns out that the last 2 equations give the same value: $a^2=17^2\implies a=.......$

Once you find a, then $b=\frac{1734}{a}\implies b=.......$

Can you finish the rest of this?

Does this make sense?

--Chris

7. Originally Posted by CaptainBlack
Do the partial derivatives, plug into the equations and then the coefficients of like powers of x and y (x and t in the second question) must be equal. That will give you a set of simultaneous equations for the unknowns independent of x and y which you should solve.

In fact for question 1, you will obtain a solution for a and b from the first of the C-R equations, and you will just need to show that this is also a solution for the second

RonL
I'm sorry but i've done some corrections now. My original post was wrong. I'm sorry for that.Thanks for your time.

8. Originally Posted by Chris L T521
I figured you were missing some sort of exponential function here.

Well, you still need to find $\frac{\partial^2u}{\partial t^2}$ and $\frac{\partial^2u}{\partial x^2}$

$\because u(x,t)=e^{x+at}(6x+bt)$

$\frac{\partial u}{\partial t}=e^{x+at}(b)+ae^{x+at}(6x+bt)=(b+6ax+abt)e^{x+at }$

$\frac{\partial^2u}{\partial t^2}=e^{x+at}(ab)+ae^{x+at}(b+6ax+abt)=(2ab+6a^2x+ a^2bt)e^{x+at}$

$\frac{\partial u}{\partial x}=e^{x+at}(6)+e^{x+at}(6x+bt)=(6+6x+bt)e^{x+at}$

$\frac{\partial^2 u}{\partial x^2}=e^{x+at}(6)+e^{x+at}(6+6x+bt)=(12+6x+bt)e^{x+ at}$

Thus,

$(2ab+6a^2x+a^2bt)e^{x+at}=17^2(12+6x+bt)e^{x+at}$

$\implies 2ab+6a^2x+a^2bt=3468+1734x+17^2bt)$

We can come up with 3 simultaneous equations equations:

$\left\{\begin{array}{lcl}
2ab & = & 3468\\
6a^2 & = & 1734 \\
a^2b & = & 17^2b \\
\end{array}
\right.
$

It turns out that the last 2 equations give the same value: $a^2=17^2\implies a=.......$

Once you find a, then $b=\frac{1734}{a}\implies b=.......$

Can you finish the rest of this?

Does this make sense?

--Chris
Thank you very much Chris.I think I can do the rest.