**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!