33–42. Solving initial value problems Solve the following initial value problems.
y'(t) = 1 + eᵗ, y(0) = 4

Verified step by step guidance

1. Identify the given differential equation and initial condition: \(y'(t) = 1 + e^{t}\) with \(y(0) = 4\).
2. Recognize that this is a first-order ordinary differential equation where \(y'(t)\) is given explicitly, so you can find \(y(t)\) by integrating the right-hand side with respect to \(t\).
3. Set up the integral to find \(y(t)\): \(y(t) = \int (1 + e^{t}) \, dt + C\), where \(C\) is the constant of integration.
4. Compute the integral: \(\int 1 \, dt = t\) and \(\int e^{t} \, dt = e^{t}\), so \(y(t) = t + e^{t} + C\).
5. Use the initial condition \(y(0) = 4\) to solve for \(C\): substitute \(t=0\) into \(y(t)\) to get \$4 = 0 + e^{0} + C\(, then solve for \)C$.

Key Concepts

Initial Value Problems (IVPs)

An initial value problem involves finding a function that satisfies a differential equation and meets a specified initial condition, such as y(0) = 4. This condition helps determine the unique solution among infinitely many possible solutions.

Solving First-Order Differential Equations

A first-order differential equation relates a function and its first derivative. Solving it often involves integrating the derivative expression to find the original function, plus a constant of integration determined by the initial condition.

Integration of Exponential Functions

Integrating expressions involving exponential functions like e^t requires applying the rule that the integral of e^t with respect to t is e^t plus a constant. This is essential for solving the given differential equation y'(t) = 1 + e^t.