Final answer:
The solution to the differential equation dy/dx = sin(x+y) + cos(x+y) can be found by separating variables and integrating. The general solution is sin(x+y) + cos(x+y) = ± e^(-x + C).
Step-by-step explanation:
The solution of the differential equation dy/dx = sin(x+y) + cos(x+y) can be found using the method of separating variables. Here are the steps:
- Start by rearranging the equation to isolate dy/dx on one side: dy = (sin(x+y) + cos(x+y))dx.
- Next, separate the variables by dividing both sides of the equation by sin(x+y) + cos(x+y): dy/(sin(x+y) + cos(x+y)) = dx.
- Integrate both sides of the equation. The integral of dy/(sin(x+y) + cos(x+y)) with respect to y gives us ln|sin(x+y) + cos(x+y)| + C1, and the integral of dx with respect to x gives us x + C2.
- Combine the constants of integration to get the general solution: ln|sin(x+y) + cos(x+y)| = -x + C.
- Finally, exponentiate both sides of the equation to eliminate the natural logarithm and solve for y: |sin(x+y) + cos(x+y)| = e^(-x + C), which simplifies to sin(x+y) + cos(x+y) = ± e^(-x + C).