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Magazine Problem 1
Solve the problem and verify your solution completely. $$ \begin{array}{ll} \frac{\partial y}{\partial x}+4 \frac{\partial y}{\partial t}=-8 t & \text { for } t>0, x>0; \\ t \rightarrow 0^{+}, y \rightarrow 0 & \text { for } x>0; \\ x \rightarrow 0^{+}, y \rightarrow 2 t^{2} & \text { for } t>0. \end{array} $$
Problem 1
Interpret and solve the problem:
$$
\begin{array}{ll}
\frac{\partial^{2} y}{\partial t^{2}}=\frac{\partial^{2} y}{\partial x^{2}} &
\text { for } t>0,0
Problem 2
Let the point with coordinates \((x, y, t)\) be in the first octant of the rectangular \(x, y, t\) space. Let that point approach the origin along the curve $$ \begin{array}{l} x^{2}=4 a^{2} t, \\ y^{2}=4 a^{2} t, \end{array} $$ in which \(a\) is positive but otherwise arbitrary. Show that as \(x, y, t \rightarrow 0^{+}\) in the manner described above, \(u\) may be made to approach any desired number between zero and unity.
Problem 2
Solve the problem and verify your solution completely. $$ \begin{array}{ll} \frac{\partial y}{\partial x}+2 \frac{\partial y}{\partial t}=4 t & \text { for } t>0, x>0; \\ t \rightarrow 0^{+}, y \rightarrow 0 & \text { for } x>0; \\ x \rightarrow 0^{+}, y \rightarrow 2 t^{3} & \text { for } t>0. \end{array} $$
Problem 5
Solve the problem and verify your solution completely. $$ \begin{array}{ll} \frac{\partial^{2} y}{\partial x^{2}}=16 \frac{\partial^{2} y}{\partial t^{2}} & \text { for } t>0, x>0 ; \\ t \rightarrow 0^{+}, y \rightarrow 0 & \text { for } x>0; \\ t \rightarrow 0^{+}, \frac{\partial y}{\partial t} \rightarrow-2 & \text { for } x>0; \\ x \rightarrow 0^{+}, y \rightarrow t & \text { for } t>0; \\ \lim _{x \rightarrow \infty} y(x, t) \text { exists } & \text { for } t>0. \end{array} $$
Problem 6
Solve the problem and verify your solution completely. $$ \begin{array}{ll} \frac{\partial^{2} y}{\partial t^{2}}=4 \frac{\partial^{2} y}{\partial x^{2}} & \text { for } t>0, x>0 ; \\ t \rightarrow 0^{+}, y \rightarrow 0 & \text { for } x>0; \\ t \rightarrow 0^{+}, \frac{\partial y}{\partial t} \rightarrow 2 & \text { for } x>0; \\ x \rightarrow 0^{+}, y \rightarrow \sin t & \text { for } t>0; \\ \lim _{x \rightarrow \infty} y(x, t) \text { exists } & \text { for } t>0. \end{array} $$
