what is the particular solution of the differential equation (x + y)dx + (x - y)dy = 0 when x=0 and y=4



The given differential equation is:

\[
(x + y) \, dx + (x - y) \, dy = 0
\]

### Step 1: Rearrange the equation

We can divide both sides by \( (x - y) \) to simplify the equation:

\[
\frac{(x + y)}{(x - y)} \, dx = -dy
\]

This gives us:

\[
\frac{(x + y)}{(x - y)} \, dx + dy = 0
\]

### Step 2: Find an integrating factor or approach separability

This form doesn't immediately lend itself to separable variables. We try solving by using an integrating factor approach or by simplifying using substitution.

### Step 3: Use substitution

Let's use the substitution \( u = x + y \). Then, differentiate \( u \):

\[
du = dx + dy
\]

Substitute into the equation:

\[
\frac{u}{(x - y)} \, dx + dy = 0
\]

This equation becomes more complex and suggests more steps ahead. Let's find a more targeted approach.

what is the particular solution of the differential equation (x + y)dx + (x - y)dy = 0 when x=0 and y=4

Solve the differential equation (x + y)dx + (x - y)dy = 0 with initial condition x=0 and y=4 using substitution and the integrating factor method. Step-by-step solution and substitution techniques for undergraduate math students.

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