2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.
y′(t) = √(y/t)

Verified step by step guidance

1. Rewrite the differential equation \(y'(t) = \sqrt{\frac{y}{t}}\) in Leibniz notation as \(\frac{dy}{dt} = \sqrt{\frac{y}{t}}\) to clearly see the relationship between \(y\) and \(t\).
2. Recognize that the equation is separable, meaning you can rearrange terms to isolate \(y\) on one side and \(t\) on the other. Start by expressing \(\frac{dy}{dt} = \frac{\sqrt{y}}{\sqrt{t}}\).
3. Separate variables by writing \(\frac{dy}{\sqrt{y}} = \frac{dt}{\sqrt{t}}\). This allows you to integrate both sides with respect to their own variables.
4. Integrate both sides: \(\int \frac{1}{\sqrt{y}} \, dy = \int \frac{1}{\sqrt{t}} \, dt\). Recall that \(\int y^{-1/2} dy = 2\sqrt{y} + C\) and similarly for \(t\).
5. After integrating, solve the resulting equation for \(y\) to express the general solution explicitly in terms of \(t\) and an arbitrary constant of integration.

Key Concepts

Separable Differential Equations

A separable differential equation can be written as a product of a function of t and a function of y, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the general solution.

Integration Techniques

Solving separable equations requires integrating both sides after separation. Familiarity with basic integration rules, including power functions and substitution, is essential to evaluate the integrals and express the solution explicitly or implicitly.

Initial Conditions and General Solutions

The general solution of a differential equation includes an arbitrary constant representing a family of solutions. Understanding how to interpret and apply initial conditions helps specify a unique solution from this family.