Partial Differential Equations

Assignment 2

1. [15 points] Consider the PDE$Lu:=x(u_x)^2 + y u_y - u=0$.

   1. Find the equation of the Monge's cone at$(1,1,-1)$.

   2. Is it possible to find Monge's cone at$(0,0,0)$? Conclude the solvability of the PDE with initial data given in a curve containing origin.

   3. Find the integral surface passing through the line$y = 1, \; x + z = 0$.

2. [30 points] Solve the following IVP:

   1. $u_y = u_x^3$,$u(x,0)=2x^{3/2}$.

   2. $u_x^2+u_y^2 = 1,u(x, y) = 0$on the line$x + y = 1$.

   3. $xp^2+yq^2 = z,y=1$on the line$x + z = 0$.

   4. $u_t + (x \cos t) u_x = 0, \; u(x, 0) = \frac{1}{1 + x^2}, \; x \in \mathbb{R}, \; t > 0$.

   5. $u_t + (x + t)u_x + t(x + 1)u = 0, \; u(x, 0) = \phi(x), \; x \in \mathbb{R}, \; t > 0$.

   6. $u_t + u^2 u_x = 0, \; u(x, 0) = x, \; x \in \mathbb{R}, \; t > 0$.

3. [10 points] Solve:

   1. $u_t - \sqrt{u_{x_1}^2 + u_{x_2}^2} = 0, \; u(x_1, x_2, t_0) = \psi(x_1^2 + x_2^2), \; \psi' > 0; \; (x_1, x_2) \in \mathbb{R}^2, \; t > t_0$

   2. $u_t + u(u_x+u_y) = 0,\,x,y \in \mathbb{R},\,t>0, u(x,y,0)=x+y,\,x,y \in \mathbb{R}.$

4. [10 points] Solve:

   $$u_t + \frac{1}{1 + |x|}u_x = 0, \; u(x, 0) = \phi(x), \; x \in \mathbb{R}, \; t > 0.$$