Author Topic: Home Assignment 1 Solutions  (Read 897 times)

Emily Deibert

  • APM346
  • Elder Member
  • *
  • Posts: 100
  • Karma: 0
    • View Profile
Home Assignment 1 Solutions
« on: September 26, 2015, 08:29:51 PM »
EDIT: For clarity, I have created separate topics for each of the below problems.


Hello,

Here are my solutions to Home Assignment 1. Please comment with any mistakes or typos you notice!

1.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.1
a) Linear homogeneous
    b) Nonlinear (quasilinear) homogeneous
    c) Linear homogeneous
    d) Nonlinear (quasilinear) inhomogeneous
    e) Nonlinear homogeneous (though I am not sure, as this one seems like it could also be called inhomogeneous semilinear?)
    f) Nonlinear inhomogeneous
    g) Nonlinear inhomogeneous
    h) Nonlinear inhomogeneous
    i) Nonlinear inhomogeneous
    j) Nonlinear homogeneous

2.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.2
a) 2nd order linear homogeneous
    b) 2nd order nonlinear (quasilinear) homogeneous
    c) 3rd order linear homogeneous
    d) 3rd order nonlinear (quasilinear) homogeneous
    e) 4th order linear homogeneous
    f) 4th order linear homogeneous
    g) 4th order linear inhomogeneous
    h) 4th order linear homogeneous

3.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.3
a) \begin{equation}
u = \int f(x)dx + g(y)
\end{equation}
    b) \begin{equation}
u = (Ce^{2y})x + f(y)
\end{equation}
Incorrect. V.I.
    c) \begin{equation}
u = e^{x+y} + \int f(x)dx + g(y)
\end{equation}
    d) I am not very confident in this answer, so hopefully someone is able to explain this question to me! \begin{equation}
u = \frac{e^xe^y}{(1-2y)}
\end{equation}

4.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.4
a) \begin{equation}
u = e^{x + g(y)}
\end{equation}
    b) \begin{equation}
u = e^{e^2x + g(y)}
\end{equation}
    c) I am a little confused about this question, so I don't have an answer yet. Below is my process so far: \begin{equation}
u_{xy} = u_{y}u_{x} \Rightarrow
\frac{u_{xy}}{u_{x}} = u_{y} \Rightarrow
ln(u_{x}) = u_{y} \Rightarrow
\end{equation}

5.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.5
a) \begin{equation}
u = y\int\Bigg(\int f(x)dx\Bigg)dx + \int\Bigg(\int g(x)dx\Bigg)dx + xh(y) + j(y)
\end{equation}
    b) \begin{equation}
u = \int \int f(x,y)dydz + \int g(x,z)dx + h(y,z)
\end{equation}
    c) \begin{equation}
u = sin(x)sin(y) + y\int \Bigg(\int f(x)dx \Bigg)dx + \int \Bigg(\int g(x)dx\Bigg)dx + xh(y) + j(y)
\end{equation}
    d) \begin{equation}
u = -cos(x)cos(y)cos(z) + \int \int f(x,y)dydx + \int g(x,z)dx + h(y,z)
\end{equation}
    e) \begin{equation}
u = -yzcos(x) - xzcos(y) - xycos(z) + \int \int f(x,y)dydx + \int g(x,z)dx + h(y,z)
\end{equation}

Also, I am not very good with Latex so if anyone has any suggestions on how to improve any of my formatting please let me know. Thank you!
« Last Edit: October 03, 2015, 04:49:20 AM by Victor Ivrii »

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 1332
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Home Assignment 1 Solutions
« Reply #1 on: September 27, 2015, 02:32:12 PM »
Emily, please don't create bloody mess next time: it would be a good idea to have for each problem (I mean problems, not their subproblems) separate topics and also to provide links (I did it for you).

It looks like things go well.

About $\LaTeX$: see
http://forum.math.toronto.edu/index.php?topic=610.0
In particular, I strongly recommend books of George Grätzer

So far my only grudge is: don't write cos,  write \cos (with the space after), then it will be upright and the proper horizontal spacing will be provided (sure, the same for all operators

Also see
http://forum.math.toronto.edu/index.php?board=8.0

In the online textbook you can see html sources to see how I did some math snippets.



Emily Deibert

  • APM346
  • Elder Member
  • *
  • Posts: 100
  • Karma: 0
    • View Profile
Re: Home Assignment 1 Solutions
« Reply #2 on: September 29, 2015, 03:23:56 PM »
Oops, sorry Professor! For future home assignments I will post in separate topics. Thank you for the tips on Latex formatting!

Zaihao Zhou

  • APM346
  • Full Member
  • *
  • Posts: 29
  • Karma: 0
    • View Profile
Re: Home Assignment 1 Solutions
« Reply #3 on: September 29, 2015, 07:40:59 PM »
Hello,

Here are my solutions to Home Assignment 1. Please comment with any mistakes or typos you notice!

1.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.1
a) Linear homogeneous
    b) Nonlinear (quasilinear) homogeneous
    c) Linear homogeneous
    d) Nonlinear (quasilinear) inhomogeneous
    e) Nonlinear homogeneous (though I am not sure, as this one seems like it could also be called inhomogeneous semilinear?)
    f) Nonlinear inhomogeneous
    g) Nonlinear inhomogeneous
    h) Nonlinear inhomogeneous
    i) Nonlinear inhomogeneous
    j) Nonlinear homogeneous

2.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.2
a) 2nd order linear homogeneous
    b) 2nd order nonlinear (quasilinear) homogeneous
    c) 3rd order linear homogeneous
    d) 3rd order nonlinear (quasilinear) homogeneous
    e) 4th order linear homogeneous
    f) 4th order linear homogeneous
    g) 4th order linear inhomogeneous
    h) 4th order linear homogeneous

3.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.3
a) \begin{equation}
u = \int f(x)dx + g(y)
\end{equation}
    b) \begin{equation}
u = (Ce^{2y})x + f(y)
\end{equation}
    c) \begin{equation}
u = e^{x+y} + \int f(x)dx + g(y)
\end{equation}
    d) I am not very confident in this answer, so hopefully someone is able to explain this question to me! \begin{equation}
u = \frac{e^xe^y}{(1-2y)}
\end{equation}

4.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.4
a) \begin{equation}
u = e^{x + g(y)}
\end{equation}
    b) \begin{equation}
u = e^{e^2x + g(y)}
\end{equation}
    c) I am a little confused about this question, so I don't have an answer yet. Below is my process so far: \begin{equation}
u_{xy} = u_{y}u_{x} \Rightarrow
\frac{u_{xy}}{u_{x}} = u_{y} \Rightarrow
ln(u_{x}) = u_{y} \Rightarrow
\end{equation}

5.
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter1/S1.P.html#problem-1.P.5
a) \begin{equation}
u = y\int\Bigg(\int f(x)dx\Bigg)dx + \int\Bigg(\int g(x)dx\Bigg)dx + xh(y) + j(y)
\end{equation}
    b) \begin{equation}
u = \int \int f(x,y)dydz + \int g(x,z)dx + h(y,z)
\end{equation}
    c) \begin{equation}
u = sin(x)sin(y) + y\int \Bigg(\int f(x)dx \Bigg)dx + \int \Bigg(\int g(x)dx\Bigg)dx + xh(y) + j(y)
\end{equation}
    d) \begin{equation}
u = -cos(x)cos(y)cos(z) + \int \int f(x,y)dydx + \int g(x,z)dx + h(y,z)
\end{equation}
    e) \begin{equation}
u = -yzcos(x) - xzcos(y) - xycos(z) + \int \int f(x,y)dydx + \int g(x,z)dx + h(y,z)
\end{equation}

Also, I am not very good with Latex so if anyone has any suggestions on how to improve any of my formatting please let me know. Thank you!

For 1(h), the operation  \begin{equation}\ L = x\partial_{x}+y\partial_{y}+z\partial_{z} \end{equation} is a linear operation, you can verify by L(a+b) = L(a) + L(b) and L(ca) = cL(a). So i think it is linear homo.