First-order partial differential equations
General · Mathematics
Study notes
Q: Solve u_x+uu_y = 0, u(0,y) = y (Burgers). Characteristics: dx/1 = dy/u = du/0. du = 0: u = const along characteristics! dy/dx = u: lines y = ux+c. Initial: u(0,y₀) = y₀: characteristic: y = y₀x+y₀... wait: y = ux+y₀, u = y₀. Implicit: u = y-ux: u = y/(1+x)! Check: u_x = -y/(1+x)², u_y = 1/(1+x): u_x+uu_y = 0 ✓! (Shock at x = -1!)