42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.
a. Find the general solution of the equation.


y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1

Verified step by step guidance

1. Identify that the given differential equation is separable: \(y'(t) = \frac{t^2}{y^2 + 1}\).
2. Rewrite the equation in separable form by expressing \(y'\) as \(\frac{dy}{dt}\) and rearranging terms to isolate \(y\) and \(t\): \( (y^2 + 1) dy = t^2 dt\).
3. Integrate both sides separately: \(\int (y^2 + 1) dy = \int t^2 dt\).
4. Compute the integrals: The left side becomes \(\int y^2 dy + \int 1 dy = \frac{y^3}{3} + y\), and the right side becomes \(\frac{t^3}{3} + C\), where \(C\) is the constant of integration.
5. Write the implicit general solution as \(\frac{y^3}{3} + y = \frac{t^3}{3} + C\). Use the initial conditions \(y(-1) = 1\), \(y(0) = 0\), and \(y(-1) = -1\) to solve for the constant \(C\) for each case.

Key Concepts

Separable Differential Equations

A separable differential equation can be written as the 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.

Implicit Solutions

An implicit solution is a relation involving both variables that satisfies the differential equation but is not explicitly solved for y. Often, after integrating, the solution is given implicitly, requiring further manipulation or interpretation to understand the behavior of y.

Initial Conditions and Solution Curves

Initial conditions specify particular values of y at given t, allowing determination of the constant of integration and thus a unique solution curve. The direction field or slope field graphically represents these solutions, showing how different initial values lead to different solution trajectories.