Differential Equations Answers

Solve the equation xu_x + (x+y)u_y = 1. with the initial condition u(1,y) = y. Is the solution defined everywhere?

Show that there exists a unique solution for the system:

U_x = 3x^2y+y
U_y = x^3+x

together with the initial condition u(0,0) = 0.

Consider the equation yu_x - xu_y = 0 (y>0). Check for each of the following initial conditions whether the problem is solvable. If it is solvable, find a solution. If it is not, explain why:
(a) u(x,0) = x^2
(b) u(x,0) = x
(c) u(x,0) = x, x>0

Show that each of the following equations has a solution of the form u(x,y) = e^(ax+by). Find the constants a,b for each example:

a) u_x + 3u_y + u = 0.

b) u_xx + u_yy = 5e^(x-2y).

If my work says use cross products to solve each proportion what I'm susposed to do?

Show that the following equation has a solution of the form u(x,y) = e^(ax+by) and find the constants a and b:

U_xx + U_yy = 5e^(x-2y).

Show that the following equation has a solution of the form u(x,y) = e^(ax + by), and find the constants a,b:

U_x + 3U_y + U =0.

limits as x approaches -infinity (e^(2x)+e^x)/(ae^(2x)-e^x)

Determine the value of tan⁡〖45°-tan⁡〖30°〗 〗/(1-tan⁡〖45° .tan⁡〖30°〗 〗 ) .

y^2dx=(x^3-xy)dy