Key Concepts and Formulas

• Exact Differential Equations: An equation of the form $M(x, y)dx + N(x, y)dy = 0$ is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. The solution is given by $\int M dx + \int (\text{terms in N not containing x}) dy = C$.
• Differential of a Quotient: $d\left(\frac{u}{v}\right) = \frac{v \, du - u \, dv}{v^2}$. In particular, $d\left(\frac{x}{y}\right) = \frac{y \, dx - x \, dy}{y^2}$ and $d\left(\frac{y}{x}\right) = \frac{x \, dy - y \, dx}{x^2}$.
• Integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.

Step-by-Step Solution

Step 1: Rearrange the Differential Equation

We are given the differential equation $y(1+xy) dx = x \, dy$. Our goal is to rearrange this equation to a form that resembles a known exact differential or can be easily integrated.

Expanding the left side, we get: $y \, dx + xy^2 \, dx = x \, dy$

Now, we want to isolate terms involving $x \, dy$ and $y \, dx$ on one side. Rearranging the terms: $xy^2 \, dx = x \, dy - y \, dx$ Explanation: The rearrangement aims to group the $x \, dy$ and $y \, dx$ terms together as it suggests the possibility of forming a differential of a quotient.

Step 2: Transform into a Standard Exact Differential Form

We have $xy^2 \, dx = x \, dy - y \, dx$. We want to manipulate this equation to resemble $d\left(\frac{x}{y}\right) = \frac{y \, dx - x \, dy}{y^2}$. Notice that the right-hand side is the negative of the numerator of $d\left(\frac{x}{y}\right)$.

To get the $y^2$ term in the denominator on the right-hand side, we divide the entire equation by $y^2$: $\frac{xy^2 \, dx}{y^2} = \frac{x \, dy - y \, dx}{y^2}$

Simplifying the left side: $x \, dx = \frac{x \, dy - y \, dx}{y^2}$

Now, we can rewrite the right-hand side using the differential of a quotient: $x \, dx = -\left(\frac{y \, dx - x \, dy}{y^2}\right) = -d\left(\frac{x}{y}\right)$

So, our differential equation becomes: $x \, dx = -d\left(\frac{x}{y}\right)$ Explanation: Dividing by $y^2$ and recognizing the differential of the quotient is a key step. The correct identification of the negative sign is crucial for the subsequent integration.

Step 3: Integrate Both Sides

Now that the equation is in a separable form, we can integrate both sides: $\int x \, dx = \int -d\left(\frac{x}{y}\right)$

Performing the integration: $\frac{x^2}{2} = -\frac{x}{y} + C$ Explanation: Integrating both sides leads to a general solution involving an arbitrary constant of integration, $C$.

Step 4: Use the Initial Condition to Find the Constant C

The problem states that the curve passes through the point $(1, -1)$. We substitute $x=1$ and $y=-1$ into our general solution to find the value of $C$: $\frac{(1)^2}{2} = -\frac{1}{-1} + C$ $\frac{1}{2} = 1 + C$

Solving for $C$: $C = \frac{1}{2} - 1$ $C = -\frac{1}{2}$ Explanation: Using the initial condition, we can determine the specific solution to the differential equation.

Step 5: Express y as a Function of x ($f(x)$)

Substitute the value of $C = -\frac{1}{2}$ back into our general solution: $\frac{x^2}{2} = -\frac{x}{y} - \frac{1}{2}$

Isolating the term with $y$: $\frac{x}{y} = -\frac{x^2}{2} - \frac{1}{2}$

Combining the terms on the right side: $\frac{x}{y} = -\frac{x^2+1}{2}$

Taking the reciprocal of both sides: $\frac{y}{x} = -\frac{2}{x^2+1}$

Finally, multiply both sides by $x$ to get $y$ in terms of $x$: $y = -\frac{2x}{x^2+1}$ So, the function is $f(x) = -\frac{2x}{x^2+1}$. Explanation: Algebraic manipulation is used to isolate $y$ and obtain the function $f(x)$.

Step 6: Calculate $f\left(-\frac{1}{2}\right)$

The question asks for the value of $f(x)$ when $x = -\frac{1}{2}$. We substitute this value into our derived function $f(x)$: $f\left(-\frac{1}{2}\right) = -\frac{2\left(-\frac{1}{2}\right)}{\left(-\frac{1}{2}\right)^2+1}$

Simplify the expression: $f\left(-\frac{1}{2}\right) = -\frac{-1}{\frac{1}{4}+1}$ $f\left(-\frac{1}{2}\right) = \frac{1}{\frac{1}{4}+\frac{4}{4}}$ $f\left(-\frac{1}{2}\right) = \frac{1}{\frac{5}{4}}$ $f\left(-\frac{1}{2}\right) = \frac{4}{5}$ Explanation: Finally, we evaluate the function $f(x)$ at $x = -\frac{1}{2}$ to obtain the answer.

Common Mistakes & Tips

• Sign Errors: Pay close attention to the negative signs, especially when manipulating the differential equation and integrating.
• Constant of Integration: Always remember to add the constant of integration, $C$, after integrating.
• Algebraic Mistakes: Be careful with algebraic manipulations when isolating $y$ and evaluating $f(x)$.

Summary

We solved the given differential equation by rearranging it into an exact differential form, specifically related to the differential of a quotient. After integrating both sides and using the initial condition to find the constant of integration, we obtained the function $f(x)$. Finally, we evaluated $f\left(-\frac{1}{2}\right)$ to get the desired result.

The final answer is \boxed{\frac{4}{5}}, which corresponds to option (B).