$$ y d x + ( y + an ( x + y ) ) d y = 0 $$

Question

$$ y d x + ( y + 	an ( x + y ) ) d y = 0 $$

UpStudy AI · Accepted Answer

We are given the differential equation

$$
y\,dx + \Bigl(y + \tan(x+y)\Bigr)\,dy = 0.
$$

**Step 1. Substitute $$ u = x+y $$**

Let

$$
u = x+y.
$$

Differentiate to find $$ du $$:

$$
du = dx + dy \quad \Longrightarrow \quad dx = du - dy.
$$

**Step 2. Rewrite the Equation**

Substitute $$ dx = du-dy $$ and $$ u = x+y $$ into the original equation:

$$
y\,(du-dy) + \left(y+\tan(u)\right)dy = 0.
$$

Distribute $$ y $$:

$$
y\,du - y\,dy + y\,dy + \tan(u)\,dy = 0.
$$

The terms $$-y\,dy$$ and $$y\,dy$$ cancel, leaving

$$
y\,du + \tan(u)\,dy = 0.
$$

**Step 3. Separate Variables**

We can rearrange the equation to separate $$ u $$ and $$ y $$:

$$
y\,du = -\tan(u)\,dy.
$$

Divide both sides by $$ \tan(u) $$ and by $$ y $$ (assuming $$ y \neq 0 $$ and $$\tan(u) \neq 0$$):

$$
\frac{du}{\tan(u)} = -\frac{dy}{y}.
$$

Since

$$
\frac{1}{\tan(u)} = \cot(u),
$$

the equation becomes

$$
\cot(u)\,du = -\frac{dy}{y}.
$$

**Step 4. Integrate Both Sides**

Integrate the left-hand side with respect to $$ u $$ and the right-hand side with respect to $$ y $$:

$$
\int \cot(u)\,du = -\int \frac{dy}{y}.
$$

Recall that

$$
\int \cot(u)\,du = \ln|\sin(u)| + C_1,
$$
$$
\int \frac{dy}{y} = \ln|y| + C_2.
$$

So we have

$$
\ln|\sin(u)| = -\ln|y| + C,
$$

where $$ C $$ absorbs any constants.

**Step 5. Simplify the Result**

Combine the logarithms:

$$
\ln|\sin(u)| + \ln|y| = C.
$$

Using the logarithm property $$ \ln a + \ln b = \ln(ab) $$, we get

$$
\ln|y\sin(u)| = C.
$$

Exponentiating both sides:

$$
|y\sin(u)| = e^C,
$$

where $$ e^C $$ is an arbitrary positive constant denoted by $$ C_1 $$. Dropping the absolute value and absorbing the sign into the constant, we write the general solution as

$$
y\sin(u) = C_1.
$$

Recall that $$ u = x+y $$, so the solution in terms of $$ x $$ and $$ y $$ is

$$
y\sin(x+y) = C.
$$

Thus, the general solution is

$$
y\sin(x+y) = C.
$$
The general solution to the differential equation $$ y\,dx + (y + \tan(x+y))\,dy = 0 $$ is $$ y\sin(x+y) = C $$, where $$ C $$ is an arbitrary constant.

\( y d x + ( y + \tan ( x + y ) ) d y = 0 \)

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