Initial Value Problems
Solve the initial value problems in Exercises 89–92.


dy/dx = (𝓍 + 1/𝓍)² , y(1)= 1

Verified step by step guidance

1. Rewrite the differential equation clearly: \(\frac{dy}{dx} = \left(x + \frac{1}{x}\right)^2\).
2. Expand the right-hand side expression: \(\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}\).
3. Set up the integral to find \(y\): \(y = \int \left(x^2 + 2 + \frac{1}{x^2}\right) \, dx + C\).
4. Integrate each term separately: \(\int x^2 \, dx\), \(\int 2 \, dx\), and \(\int \frac{1}{x^2} \, dx\).
5. Use the initial condition \(y(1) = 1\) to solve for the constant of integration \(C\) after finding the general solution.

Key Concepts

Initial Value Problems (IVPs)

An initial value problem involves solving a differential equation with a given initial condition, which specifies the value of the unknown function at a particular point. This condition allows us to find a unique solution that fits both the differential equation and the initial value.

Separable Differential Equations

A separable differential equation can be written so that all terms involving one variable are on one side and all terms involving the other variable are on the opposite side. This allows integration of both sides separately to find the general solution.

Integration of Algebraic Expressions

Solving the given differential equation requires integrating algebraic expressions, such as polynomials or rational functions. Understanding how to expand, simplify, and integrate these expressions is essential to find the explicit form of the solution.