# Homework Help: Answer check transport equation (1st order linear PDE)

1. Jun 10, 2009

### jianxu

1. The problem statement, all variables and given/known data

Hi everyone, I just wanted to double check if I've solved this correctly?

Given:
$$\left.\frac{du}{dx} + sin(x)\frac{du}{dy} = 0$$

$$\left.-\infty < x < \infty$$

y > 0

$$\left.u(\frac{\pi}{2} , y ) = y^{2}$$

Solve the PDE

2. Relevant equations
Method of characteristics

3. The attempt at a solution
Using method of characteristics, first I said:

$$\left.\frac{dy}{dx}= \frac{sin x}{1}$$

taking the integral I get
$$\left. y = -cos(x) + C$$

solving for C:
C = y + cos(x)

so now,
$$\left.u(x,y) = f(y+cos(x))$$

our initial condition gives us the following:
$$\left.u(\frac{\pi}{2},y) = y^{2}$$
so:
$$\left.u(\frac{\pi}{2},y) = f(y)$$

which means:
$$\left. f(y) = y^{2}$$

Therefore:
$$\left. u(x,y) = f(y + cos x) = (y+cos(x))^{2}$$

I think this should be right but I wanted to double check. Thanks!!!

2. Jun 10, 2009

### Staff: Mentor

Why don't you, then? (Double check, that is.) One thing you should get in the habit of doing is checking your own work. Take the partials of u(x, y) and see if they satisfy your differential equation and initial conditions. If they do, you're golden.

3. Jun 10, 2009

### jianxu

hm didn't think about that before. thanks