To solve the differential equation [tex]\( 2x + 2y \frac{dy}{dx} = 0 \)[/tex], follow these steps:
1. Simplify the equation:
[tex]\[
2x + 2y \frac{dy}{dx} = 0
\][/tex]
We can divide every term by 2 to simplify:
[tex]\[
x + y \frac{dy}{dx} = 0
\][/tex]
2. Rearrange the equation to separate the variables:
[tex]\[
y \frac{dy}{dx} = -x
\][/tex]
3. Rewrite the equation in a separable form:
[tex]\[
y \frac{dy}{dx} = -x
\][/tex]
Separate the variables [tex]\(y\)[/tex] and [tex]\(x\)[/tex]:
[tex]\[
y \, dy = -x \, dx
\][/tex]
4. Integrate both sides to find the solution:
[tex]\[
\int y \, dy = \int -x \, dx
\][/tex]
Perform the integration on both sides:
[tex]\[
\frac{y^2}{2} = -\frac{x^2}{2} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
5. Solve for [tex]\(y\)[/tex]:
Multiply both sides by 2 to simplify:
[tex]\[
y^2 = -x^2 + 2C
\][/tex]
Rearrange it to find [tex]\(y\)[/tex]:
[tex]\[
y^2 = 2C - x^2
\][/tex]
Taking the square root of both sides:
[tex]\[
y = \pm \sqrt{2C - x^2}
\][/tex]
6. Express the general form of the solution:
To present the solution in the desired form of [tex]\(f(x)\)[/tex], where the constant [tex]\(C\)[/tex] is replaced by a more general constant, say [tex]\(C_1\)[/tex]:
[tex]\[
y = \pm \sqrt{C_1 - x^2}
\][/tex]
To eliminate the [tex]\( \pm \)[/tex] and to generally define it as a function, we note that it can be written as:
[tex]\[
y(x) = C_1 - \frac{x^2}{2}
\][/tex]
Therefore, the general solution of the given differential equation [tex]\( 2x + 2y \frac{dy}{dx} = 0 \)[/tex] is:
[tex]\[
y(x) = C_1 - \frac{x^2}{2}
\][/tex]
