The differential equation presented is:

$\frac{dy}{dx} = \sin(x + y)$

This type of equation can be solved using an appropriate substitution. The given hint indicates that we should look for a substitution that simplifies the equation.

Step-by-Step Solution:

1. Substitution: Let's use the substitution $v = x + y$, so that $y = v - x$.

2. Differentiate: Now, differentiate $y = v - x$ with respect to $x$: $\frac{dy}{dx} = \frac{dv}{dx} - 1$

3. Substitute into the original equation: Substituting into the original differential equation $\frac{dy}{dx} = \sin(x + y)$, we have: $\frac{dv}{dx} - 1 = \sin(v)$ Simplifying: $\frac{dv}{dx} = \sin(v) + 1$

4. Separation of variables: Now, we can separate variables to solve for $v$: $\frac{dv}{\sin(v) + 1} = dx$

5. Integration: Integrate both sides: $\int \frac{1}{\sin(v) + 1} dv = \int dx$ The integral on the right-hand side is straightforward: $x + C$ For the left-hand side, we can use a standard substitution: $\int \frac{1}{\sin(v) + 1} dv = \ln|\tan(\frac{v}{2} + \frac{\pi}{4})|$ Thus, the solution becomes: $\ln|\tan(\frac{v}{2} + \frac{\pi}{4})| = x + C$

6. Back-substitution: Recall that $v = x + y$, so substituting $v$ back in: $\ln|\tan\left(\frac{x + y}{2} + \frac{\pi}{4}\right)| = x + C$

Thus, the general solution to the differential equation is:

$\ln|\tan\left(\frac{x + y}{2} + \frac{\pi}{4}\right)| = x + C$

Would you like more details on any step or have any questions?

Related Questions:

1. How do you handle the integration of more complex trigonometric functions?
2. What are some common substitutions for solving non-linear differential equations?
3. Can this method be extended to solve higher-order differential equations?
4. How does the choice of substitution simplify solving certain differential equations?
5. What other methods could be used to solve this type of differential equation?

Tip: Always verify your solution by differentiating it and substituting back into the original equation to ensure it satisfies the differential equation.