Initial Value Problems


Solve the initial value problems in Exercises 71–90.


d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1

Verified step by step guidance

1. Identify the given differential equation and initial conditions: \( \frac{d^{2}r}{dt^{2}} = \frac{2}{t^{3}} \), with \( \frac{dr}{dt}\bigg|_{t=1} = 1 \) and \( r(1) = 1 \).
2. Integrate the second derivative \( \frac{d^{2}r}{dt^{2}} \) with respect to \( t \) to find the first derivative \( \frac{dr}{dt} \). This means computing \( \int \frac{2}{t^{3}} \, dt \). Remember to add an integration constant \( C_1 \).
3. Use the initial condition \( \frac{dr}{dt}\big|_{t=1} = 1 \) to solve for the constant \( C_1 \) after finding the general form of \( \frac{dr}{dt} \).
4. Integrate the expression for \( \frac{dr}{dt} \) with respect to \( t \) to find \( r(t) \). Again, include a second integration constant \( C_2 \).
5. Apply the initial condition \( r(1) = 1 \) to solve for \( C_2 \), completing the solution for \( r(t) \).

Key Concepts

Second-Order Differential Equations

These are differential equations involving the second derivative of a function. Solving them requires integrating twice and applying initial conditions to find the specific solution that fits the problem.

Initial Value Problems (IVP)

An IVP specifies the value of the function and its derivatives at a particular point. This information is used to determine the constants of integration after solving the differential equation.

Integration Techniques for Variable Coefficients

When the differential equation has terms like 2/t³, integration involves handling variable coefficients carefully, often requiring substitution or direct integration of power functions.