21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
y'(t) = t lnt + 1

Verified step by step guidance

1. Recognize that the given differential equation is a first-order ordinary differential equation of the form \(y'(t) = f(t)\), where \(f(t) = t \ln t + 1\).
2. To find the general solution, integrate both sides with respect to \(t\): \(y(t) = \int (t \ln t + 1) \, dt + C\), where \(C\) is an arbitrary constant.
3. Split the integral into two parts: \(\int t \ln t \, dt\) and \(\int 1 \, dt\).
4. Use integration by parts to solve \(\int t \ln t \, dt\). Let \(u = \ln t\) and \(dv = t \, dt\), then find \(du\) and \(v\) accordingly.
5. After computing the integrals, combine the results and add the constant \(C\) to write the general solution \(y(t)\).

Key Concepts

Ordinary Differential Equations (ODEs)

An ordinary differential equation relates a function and its derivatives with respect to one variable. Solving an ODE involves finding a function that satisfies the equation, often including arbitrary constants representing a family of solutions.

Integration of Functions

To solve first-order ODEs like y'(t) = f(t), we integrate the right-hand side with respect to t. Integration reverses differentiation and introduces an arbitrary constant, representing the general solution's family.

Properties of Logarithmic Functions

Understanding the natural logarithm function ln(t) and its behavior is essential when integrating expressions like t ln(t). Techniques such as integration by parts are often used to integrate products involving logarithms.