In Exercises 5-8, show that each function is a solution of...

In Exercises $5-8,$ show that each function is a solution of the given initial value problem.
$$\begin{array}{ll}{x^{2} y^{\prime}=x y-y^{2},} & {y(e)=e \quad y=\frac{x}{\ln x}} \\ {x>1}\end{array}$$

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Key Concepts

Initial Value Problem Verification

This concept involves checking that a given function meets both the differential equation and its starting condition. It requires substituting the function and its derivative into the original equation and ensuring that the initial condition is satisfied.

First-Order Differential Equations

These are equations that involve the first derivative of the unknown function. Verifying a candidate solution for such equations typically involves differentiating the candidate function and confirming it satisfies the equation.

Differentiation Techniques

Differentiation, including rules like the quotient rule, is essential for computing the derivative of a candidate solution. Accurate differentiation is critical to ensure that the substituted derivative correctly leads to the required equality in the differential equation.

Domain Considerations

When verifying solutions, it is important to consider the domain of the function, especially when functions like the logarithm are involved. The domain must be in line with both the original differential equation and the form of the candidate solution.

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