We are tasked with finding the integral surface of the linear partial differential equation:

$u_x + 2u_y = 0$ subject to the Cauchy condition that $u = \sin(x)$ when $y = 3x + 1$.

Step 1: Write down the characteristic equations

For this linear PDE, we can use the method of characteristics. The characteristic equations are derived from the coefficients of $u_x$ and $u_y$ in the PDE.

The general characteristic equations for this equation are:

$\frac{dx}{1} = \frac{dy}{2} = \frac{du}{0}$

These equations tell us that:

1. $\frac{dx}{1} = \frac{dy}{2}$, which simplifies to $2dx = dy$, or equivalently, $dy = 2dx$.
2. $\frac{du}{0}$, which means that $u$ is constant along the characteristic curves.

Step 2: Solve the characteristic equations

From $dy = 2dx$, we can integrate to find the relationship between $x$ and $y$. Integrating both sides:

$y = 2x + C_1$ where $C_1$ is a constant of integration.

Since $u$ is constant along the characteristics, we conclude that $u$ is a function of $y - 2x$, i.e.,

$u = f(y - 2x)$

where $f$ is an arbitrary function.

Step 3: Apply the Cauchy condition

Now, we apply the Cauchy condition, which is given as $u = \sin(x)$ when $y = 3x + 1$. Substituting this into the general form of $u$:

$u = f(y - 2x)$ Substitute the condition $y = 3x + 1$ into this:

$u = f((3x + 1) - 2x) = f(x + 1)$ From the Cauchy condition, we know that $u = \sin(x)$ when $y = 3x + 1$, so:

$\sin(x) = f(x + 1)$

Thus, we have the form for $f$:

$f(z) = \sin(z - 1)$

Step 4: Final expression for $u$

Now, we substitute $f(z)$ back into the general solution for $u$:

$u = f(y - 2x) = \sin((y - 2x) - 1)$ Thus, the solution to the PDE is:

$u = \sin(y - 2x - 1)$

Final Answer:

The integral surface for the given PDE is:

$u = \sin(y - 2x - 1)$

Would you like further details or clarifications on any part of this process?

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Here are some related questions to explore further:

1. What happens if the Cauchy condition was different, say $u = \cos(x)$ on $y = 2x + 1$?
2. How would we solve this problem if the PDE had a non-linear term, such as $u_x^2 + 2u_y = 0$?
3. How can we check whether this solution satisfies the original PDE and Cauchy condition?
4. How would we approach this problem if the PDE were in three variables (e.g., $u_x + 2u_y + 3u_z = 0$)?
5. How can we generalize the method of characteristics for second-order PDEs?

Tip: When solving PDEs using the method of characteristics, remember that the solution is typically a function of the combination of variables that remain constant along the characteristic curves.