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Chapter 6: Problem 26

Solve the differential equation. $$ y^{\prime}=\ln x $$

Short Answer

Expert verified
The solution of the given differential equation \(y' = \ln(x)\) is \(y(x) = x \ln(x) - x + C\).

Step by step solution

01

Understand the Type of The Differential Equation

The given equation \(y' = \ln(x)\) is a first-order differential equation. There is only first derivative of the function \(y\), and it equals to \(\ln(x)\), a known function.
02

Applying the Integral

Since the derivative of \(y\) is given, we can get \(y(x)\) by integrating both sides. Integration of \(y' = \ln(x)\) with respect to \(x\) gives the equation \(\int y' dx = \int \ln(x) dx\).
03

Evaluate the Integral

The integral of \(\ln(x)\) is \(x \ln(x) - x\) up to a constant of integration. Evaluating the integral, we get \(y(x) = x \ln(x) - x + C\), where \(C\) is the constant of integration.

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