https://file.aiquickdraw.com/mathbot/file/1729036127215yjg7cf46.jpg

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.

Solve the given differential equation by using an appropriate substitution. The DE is of the form dy/dx = f(Ax + By + C). dy/dx = sin(x + y)

Learn how to solve the differential equation dy/dx = sin(x + y) using substitution and separation of variables. This solution includes a detailed step-by-step approach with key substitutions, integrations, and final solution expression. Ideal for undergraduate students studying calculus or differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation