5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.
(t² + 1)³yy'(t) = t(y² + 4)

Verified step by step guidance

1. Rewrite the given differential equation \((t^2 + 1)^3 y y'(t) = t (y^2 + 4)\) by expressing \(y'(t)\) as \(\frac{dy}{dt}\), so it becomes \((t^2 + 1)^3 y \frac{dy}{dt} = t (y^2 + 4)\).
2. Separate the variables by moving all terms involving \(y\) to one side and all terms involving \(t\) to the other side. This gives \(\frac{y}{y^2 + 4} dy = \frac{t}{(t^2 + 1)^3} dt\).
3. Integrate both sides: compute \(\int \frac{y}{y^2 + 4} dy\) on the left and \(\int \frac{t}{(t^2 + 1)^3} dt\) on the right.
4. For the left integral, use substitution \(u = y^2 + 4\), so \(du = 2y dy\), which simplifies the integral. For the right integral, consider substitution \(v = t^2 + 1\), so \(dv = 2t dt\), to simplify the integral.
5. After integrating both sides, include the constant of integration \(C\), then solve the resulting equation explicitly for \(y\) as a function of \(t\) to find the general solution.

Key Concepts

Separable Differential Equations

A separable differential equation can be written as a product of a function of the dependent variable and a function of the independent variable. This allows the equation to be rearranged so that all terms involving one variable are on one side and all terms involving the other variable are on the opposite side, enabling integration on both sides.

Implicit and Explicit Solutions

An implicit solution defines the dependent variable in terms of the independent variable without isolating it explicitly, while an explicit solution expresses the dependent variable directly as a function of the independent variable. The problem asks for the explicit form, so after integration, solving for the dependent variable is necessary.

Integration Techniques for Nonlinear Terms

Solving separable equations often requires integrating nonlinear expressions involving powers or polynomials. Familiarity with integration rules, substitution, and algebraic manipulation is essential to handle terms like (t² + 1)³ and y² + 4, ensuring correct integration and simplification.