Key Concepts and Formulas

• Separable Differential Equations: A differential equation of the form $\frac{dy}{dx} = g(x)h(y)$ can be solved by separating variables and integrating.
• Integration: The process of finding the antiderivative of a function.
• Initial Value Problem: Finding a particular solution to a differential equation that satisfies a given initial condition.

Step-by-Step Solution

Step 1: Separate the variables

We are given the differential equation $\frac{dy}{dx} = \frac{2y}{x \log_e x}$. To separate the variables, we divide both sides by $y$ and multiply both sides by $dx$: $\frac{dy}{y} = \frac{2}{x \log_e x} dx$ This separates the $y$ terms on the left and the $x$ terms on the right.

Step 2: Integrate both sides

Now, we integrate both sides of the equation with respect to their respective variables: $\int \frac{dy}{y} = \int \frac{2}{x \log_e x} dx$ The left side integrates to $\log_e |y|$. For the right side, let $u = \log_e x$. Then, $\frac{du}{dx} = \frac{1}{x}$, so $du = \frac{1}{x} dx$. Thus, the integral becomes: $\int \frac{2}{x \log_e x} dx = \int \frac{2}{u} du = 2 \int \frac{1}{u} du = 2 \log_e |u| + C = 2 \log_e |\log_e x| + C$ Therefore, we have: $\log_e |y| = 2 \log_e |\log_e x| + C$

Step 3: Simplify the equation

Using logarithm properties, we can rewrite the equation as: $\log_e |y| = \log_e (\log_e x)^2 + C$ Exponentiating both sides with base $e$, we get: $|y| = e^{\log_e (\log_e x)^2 + C} = e^{\log_e (\log_e x)^2} \cdot e^C = (\log_e x)^2 \cdot e^C$ Let $A = e^C$, where $A$ is a positive constant. Then, we can write $y = A (\log_e x)^2$

Step 4: Apply the initial condition

We are given that the curve passes through the point $(2, (\log_e 2)^2)$. Substituting $x = 2$ and $y = (\log_e 2)^2$ into the equation, we get: $(\log_e 2)^2 = A (\log_e 2)^2$ Dividing both sides by $(\log_e 2)^2$, we find $A = 1$. Therefore, the particular solution is: $y = (\log_e x)^2$

Step 5: Find f(e)

We want to find the value of $f(e)$. Substituting $x = e$ into the equation $y = (\log_e x)^2$, we get: $f(e) = (\log_e e)^2 = (1)^2 = 1$ However, the given answer is 2. Let's re-examine Step 3.

Step 3 (Revised): Simplify the equation

We have: $\log_e |y| = 2 \log_e |\log_e x| + C$ $\log_e |y| = \log_e (\log_e x)^2 + C$ Exponentiating both sides with base $e$, we get: $|y| = e^{\log_e (\log_e x)^2 + C} = e^{\log_e (\log_e x)^2} \cdot e^C = (\log_e x)^2 \cdot e^C$ Let $A = e^C$, where $A$ is a positive constant. Then, we can write $y = A (\log_e x)^2$

Step 4 (Revised): Apply the initial condition

We are given that the curve passes through the point $(2, (\log_e 2)^2)$. Substituting $x = 2$ and $y = (\log_e 2)^2$ into the equation, we get: $(\log_e 2)^2 = A (\log_e 2)^2$ Dividing both sides by $(\log_e 2)^2$, we find $A = 1$. Therefore, the particular solution is: $y = (\log_e x)^2$

Step 5 (Revised): Find f(e)

I realize my mistake. The original equation was $\frac{dy}{dx} = \frac{2y}{x \log_e x}$. Let's review the integration in step 2. $\int \frac{dy}{y} = \int \frac{2}{x \log_e x} dx$ The left side integrates to $\log_e |y|$. For the right side, let $u = \log_e x$. Then, $\frac{du}{dx} = \frac{1}{x}$, so $du = \frac{1}{x} dx$. Thus, the integral becomes: $\int \frac{2}{x \log_e x} dx = \int \frac{2}{u} du = 2 \int \frac{1}{u} du = 2 \log_e |u| + C = 2 \log_e |\log_e x| + C$ Therefore, we have: $\log_e |y| = 2 \log_e |\log_e x| + C$ Using logarithm properties, we can rewrite the equation as: $\log_e |y| = \log_e (\log_e x)^2 + \log_e K$ (Where C = log K) $\log_e |y| = \log_e (K(\log_e x)^2)$ Exponentiating both sides with base $e$, we get: $y = K (\log_e x)^2$

Using the point (2, $(\log_e 2)^2$) $(\log_e 2)^2 = K (\log_e 2)^2$ So K = 1. However, this still gives the wrong answer. The correct answer is 2, so we need to look at the separation again.

Let's revisit Step 2 and onward. $\int \frac{dy}{y} = \int \frac{2}{x \ln x} dx$ $\ln |y| = 2 \ln |\ln x| + C$ $\ln |y| = \ln (\ln x)^2 + C$ $e^{\ln |y|} = e^{\ln (\ln x)^2 + C}$ $y = e^C (\ln x)^2$ Let $A = e^C$. Then $y = A (\ln x)^2$. Using the point $(2, (\ln 2)^2)$, $(\ln 2)^2 = A (\ln 2)^2$ So $A = 1$. Thus, $y = (\ln x)^2$. We want f(e). $f(e) = (\ln e)^2 = 1^2 = 1$

I am still getting 1. Let us re-examine the separation of variables. $\frac{dy}{dx} = \frac{2y}{x \ln x}$ $\frac{dy}{y} = \frac{2}{x \ln x} dx$ $\int \frac{dy}{y} = \int \frac{2}{x \ln x} dx$ $\ln |y| = 2 \ln |\ln x| + C$ $e^{\ln |y|} = e^{2 \ln |\ln x| + C}$ $|y| = e^{2 \ln |\ln x|} e^C$ $y = A (\ln x)^2$ where A is a constant. Given the point $(2, (\ln 2)^2)$, we have $(\ln 2)^2 = A (\ln 2)^2$ $A = 1$ Therefore, $y = (\ln x)^2$. Then $f(e) = (\ln e)^2 = 1^2 = 1$. I apologize. I am unable to arrive at the correct answer of 2.

Common Mistakes & Tips

• Always remember the constant of integration when performing indefinite integrals.
• Double-check the integration and algebraic manipulations to avoid errors.
• Be careful with the properties of logarithms and exponentials.

Summary

We solved the separable differential equation $\frac{dy}{dx} = \frac{2y}{x \log_e x}$ by separating variables, integrating both sides, and applying the initial condition $(2, (\log_e 2)^2)$. The resulting function is $y=(\log_e x)^2$. Evaluating this function at $x=e$ yields $f(e) = 1$.

Final Answer

The final answer is \boxed{1}.