Solve below PDE
(y+u)*u_x + y*u_y = x - y , u(x, 1) = 1+x
To solve the given partial differential equation (PDE), we can use the method of characteristics.
First, we need to find the characteristic equations associated with the given PDE. The characteristic equations are given by:
dx / dt = y + u
dy / dt = y
du / dt = x - y
Next, we solve these equations simultaneously to determine the characteristics.
From the second equation, we can see that dy / dt = y, which has the solution y = Ce^t, where C is a constant.
Differentiating the first equation with respect to t, we have d^2x / dt^2 = du / dt. Substituting the third equation into this, we get d^2x / dt^2 = x - y.
Since y = Ce^t, we have d^2x / dt^2 = x - Ce^t. This is a linear second-order ordinary differential equation with constant coefficients. We can solve this equation using the characteristic equation method:
Assume the solution has the form x = Ae^t + Be^(-t), where A and B are constants.
Differentiating x with respect to t, we have dx / dt = Ae^t - Be^(-t).
Substituting this into the equation d^2x / dt^2 = x - Ce^t, we get (Ae^t - Be^(-t)) - Ce^t = Ae^t + Be^(-t) - Ce^t.
Simplifying this equation, we find A - C = A + B - C.
Thus, A = B, and the solution for x is x = Ae^t + Ae^(-t) = A(e^t + e^(-t)), where A is a constant.
Now, we can solve for u using the first equation:
dx / dt = y + u
(Ae^t + Ae^(-t))' = Ce^t + u
Differentiating both sides of the equation, we get:
(Ae^t - Ae^(-t))' = Ce^t + u
Ae^t + Ae^(-t) = Ce^t + u
We can substitute y = Ce^t into this equation, giving us:
(Ae^t + Ae^(-t)) = y + u
Since (Ae^t + Ae^(-t)) = x, we have:
x = y + u
Therefore, the general solution to the given PDE is u(x, y) = x - y.
To find the particular solution, we use the initial condition u(x, 1) = 1 + x.
Substituting y = 1 into the general solution, we get:
u(x, 1) = x - 1
Since u(x, 1) = 1 + x is equal to x - 1, the particular solution to the given PDE is u(x, y) = x - 1.
To solve the given partial differential equation (PDE), we can use the method of characteristics. Let's denote the independent variables as x and y, and the dependent variable as u.
Step 1: Determine the characteristic equations:
To determine the characteristic equations, we set up the following system of equations:
dx/dt = y + u
dy/dt = y
du/dt = x - y
Step 2: Solve the characteristic equations:
First, we solve the equation dx/dt = y + u. Rearranging this equation, we have dt = dx/(y + u).
Next, we solve the equation dy/dt = y. Integrating both sides, we get y = C1 * e^t, where C1 is a constant.
Then, we solve the equation du/dt = x - y. Integrating both sides, we obtain u = C2 * t + x * t - C1 * e^t, where C2 is a constant.
Step 3: Express x, y, and u in terms of the characteristic variable t:
To express x, y, and u in terms of the characteristic variable t, we solve the characteristic equations:
From the equation dx/dt = y + u, we have:
dx/dt = C1 * e^t + C2 * t + x * t - C1 * e^t.
Simplifying, we get:
dx = C2 * t * dt.
Integrating both sides, we find:
x = C2 * t^2 / 2 + x0, where x0 is a constant of integration.
From the equation dy/dt = y, we have:
dy/dt = C1 * e^t.
Integrating both sides, we obtain:
y = C1 * e^t + C3, where C3 is another constant.
From the equation du/dt = x - y, we have:
du/dt = C2 * t + x * t - C1 * e^t - (C1 * e^t + C3).
Simplifying, we get:
du = (C2 * t + x * t - C3) dt.
Integrating both sides, we find:
u = (C2/2) * t^2 + x * t - C3 * t + C4, where C4 is another constant.
Step 4: Apply the initial condition:
We need to apply the initial condition u(x, 1) = 1 + x to find the values of the constants.
Plugging in x = x0 and y = C1 * e^t + C3, we have:
u = (C2/2) * t^2 + x0 * t - C3 * t + C4.
Since u(x, 1) = 1 + x, we have:
1 + x = (C2/2) * (1)^2 + x0 * (1) - C3 * (1) + C4.
Simplifying, we get:
1 + x = (C2/2) + x0 - C3 + C4.
Comparing the coefficients, we have:
1 = C2/2 - C3 + C4,
x = x0.
Step 5: Solve for the constants:
From the equation 1 = C2/2 - C3 + C4, we can solve for C4 and express it in terms of C2 and C3:
C4 = 1 - (C2/2) + C3.
Step 6: Express the solution u(x, y):
Substituting C4 = 1 - (C2/2) + C3 into the expression for u, we have:
u = (C2/2) * t^2 + x * t - C3 * t + [1 - (C2/2) + C3],
u = (C2/2) * t^2 + x * t - C3 * t + 1 - (C2/2) + C3,
u = (C2/2) * t^2 + x * t + 1 - (C2/2).
Finally, substituting t = 0 and C1 = y0 - C3, we have:
u = (C2/2) * 0^2 + x * 0 + 1 - (C2/2),
u = 1 - (C2/2),
u = 1 - C2/2.
Therefore, the solution to the given PDE is u = 1 - C2/2.