91Ó°ÊÓ

Integration is a fundamental process in solving differential equations. Here, we integrate both sides of an equation to find a general solution. In this context, we're dealing with a separable differential equation, where integration is applied to expressions involving separated variables. For instance, once our original equation \(\frac{dy}{dx} = \frac{y(\ln(y) - \ln(x) + 1)}{x}\) is rewritten, both sides can be integrated separately.

• Left Side: \(\int \frac{1}{y(1 + \ln(y) - \ln(x))} \,dy\)
• Right Side: \(\int \frac{1}{x} \,dx\)

Performing these integrations may require different techniques. The integral on the right is straightforward, resulting in the natural logarithm \(\ln(x)\). The one on the left involves considering function inverses or substitution methods. This can be more complex, often necessitating clever algebra to simplify the fraction.
Through carefully performed integrations, we find \(- \ln(\ln(y) - \ln(x) + 1) = \ln(x) + C_1\), with \(C_1\) being an integration constant. This formula is integral to further solving for \(y\) in the equation, showcasing the power of integration techniques.

A differential equation solution involves finding an equation describing a function, such as \(y=f(x)\), that satisfies the differential relationship. The equation given is separable, meaning it can be rearranged so that all terms involving \(y\) and \(dy\) are on one side and all terms involving \(x\) and \(dx\) are on the other. This property makes the integration process feasible and systematic.
The general solution for this separable differential equation was derived by first separating the variables:

• \(\frac{dy}{y(\ln(y) - \ln(x) + 1)} = \frac{dx}{x}\)

Next, integrating both sides presented a solution containing a constant of integration, depicted logically as
\(- \ln(\ln(y) - \ln(x) + 1) = \ln(x) + C_1\).

When it comes to finding particular solutions, initial conditions might be used. This constant \(C_1\) would then be resolved to ensure the solution meets specific conditions. These solutions are invaluable in fields like physics and engineering, allowing for modeling and prediction of real-world systems.

Isolation of variables is an essential step in solving differential equations, especially separable ones. This process involves rearranging the equation so that each type of variable is grouped together, typically making one side dependent on \(x\) and the other on \(y\). For the given differential equation \(\frac{d y}{d x}=\frac{y(\ln y-\ln x+1)}{x}\), we accomplish this in Step 1: \(\frac{dy}{y(\ln(y) - \ln(x) + 1)} = \frac{dx}{x}\).

Isolating variables allows each side of the equation to be integrated separately. This clarity is why isolating variables is a fundamental technique when dealing with differential equations. After the integration, the challenge is often to rearrange the result to solve for the dependent variable \(y\).
Through exponential techniques and some algebraic manipulation, we arrive at the final form of the solution:

• From \(\ln(y) = \ln(x) + e^{- \ln(x) - C_1} - 1\) to \(y = x \cdot e^{(e^{-\ln(x) - C_1} - 1)}\)

This final step ensures that the function \(y\) is explicitly expressed in terms of \(x\). The isolation and manipulation techniques alleviate potential complexity, making the solution more accessible. It exemplifies how solving differential equations is both analytical and methodical.