91Ó°ÊÓ

To solve a separable differential equation, we first need to separate the variables. This means getting all the terms involving the independent variable (here, x) on one side and all the terms involving the dependent variable (here, y) on the other side.

Starting with the given equation: \ \[ \frac{dy}{dx} = \frac{1}{y^2 + x} \]
We aim to move all y-terms to one side of the equation and all x-terms to the other. Multiply both sides by \[ y^2 + x \], yielding:

\[ (y^2 + x) \frac{dy}{dx} = 1 \].
Next, we rearrange and separate the variables, which can be tricky. However, representing the equation in this form helps in the next step:

\[ y^2 dy = \frac{1}{dx} - x dx \]

Recall our goal: isolate the variables y and x onto different sides of the equation.

After separating the variables, the next step involves integrating both sides with respect to their respective variables.

Given our modified equation:

\[ y^2 dy = \frac{1}{dx} - x dx \]

We notice integration symbols: \(\text{∫}\frac{1}{y^2} dy - \text{∫} x dx\)
Now, separately integrate each side:

For the left side: \[ \text{∫}\frac{1}{y^2} dy = -\frac{1}{y} \]
This is obtained using the power rule. For the right side: \[ - \text{∫} x dx = -\frac{x^2}{2} + C. \]
Here, C is the constant of integration.

Having integrated both sides, the solution contains y and x mixed with the constant C:

\[ -\frac{1}{y} = \frac{x^2}{2} + C \. \]
To isolate y, take the reciprocal and solve:

\[ y = -\frac{1}{\frac{x^2}{2} + C}\]

Rewriting this solution allows us to express y explicitly in terms of x.

• This is the general solution of the given differential equation.
• Remember, simplification and correct algebraic manipulation are critical.