Solving a Differential Equation to Find a Differentiable Function

The given problem involves finding a differentiable function that satisfies certain conditions. This can be approached as a first-order ordinary differential equation (ODE). Let's rewrite and solve the given problem to find the solution.

Understanding the Problem

We are given a differentiable function ( f ) such that:

f(e) -1 f'(x) ≠ 0 for x ≥ 1 frac{f(x)}{f(x)}x ln x 0 for x ≥ 1 f(e) -1

The last condition f(e) -1 is redundant as it is already mentioned in the first condition. The third condition is also redundant for the same reason. Therefore, we are left with the first and second conditions, and the task is to find a function ( f(x) ) that satisfies these conditions.

Rewriting the Problem in Terms of a First-Order ODE

We need to convert the given expression into a standard first-order ODE. The given expression is:

frac{1}{f(x)}f'(x) cdot x ln x 0

To make this an equation in terms of the derivative, we can isolate the derivative term:

frac{1}{f(x)} f'(x) -frac{1}{x ln x}

This can be simplified to:

frac{1}{f(x)} frac{df}{dx} -frac{1}{x ln x}

Multiplying both sides by df and dx we get:

frac{1}{f(x)} df -frac{1}{x ln x} dx

Integrating Both Sides

Now, we integrate both sides of the equation:

int frac{1}{f(x)} df -int frac{1}{x ln x} dx

The integral on the left side is:

log |f(x)| C_1

The integral on the right side is:

-log |ln x| C_2

Combining the constants of integration, we get:

log |f(x)| -log |ln x| C

This can be further simplified to:

log |f(x)| -log |ln x| log C

Using the properties of logarithms, we can write:

log |f(x)| log left(frac{C}{ln x}right)

Exponentiating both sides, we get:

|f(x)| frac{C}{ln x}

Since f(x) is a differentiable function, we can drop the absolute value:

f(x) frac{C}{ln x}

Applying Boundary Conditions

Now, we apply the boundary condition f(e) -1 to find the constant C.

f(e) frac{C}{ln e} -1

Since ln e 1, we get:

C -1

Therefore, the function becomes:

f(x) frac{-1}{ln x} -frac{1}{ln x}

Conclusion

The differentiable function that satisfies the given conditions is:

f(x) -frac{1}{ln x}

Additional Insights

Understanding how to work with differential equations and logarithmic functions is crucial in many areas of mathematics, physics, and engineering. The process involves recognizing patterns and applying integration techniques.

References

For further reading and understanding of differential equations, the following references are recommended:

First-Order Linear Differential Equations Logistic Growth Differential Equations Solutions of Linear Differential Equations of First Order