In Exercises 125–128 solve the differential equation.
127. yy' = sec(y²)sec²(x)

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1. Rewrite the given differential equation \(yy' = \sec(y^{2}) \sec^{2}(x)\) by expressing \(y'\) as \(\frac{dy}{dx}\). This gives \(y \frac{dy}{dx} = \sec(y^{2}) \sec^{2}(x)\).
2. Separate the variables so that all terms involving \(y\) are on one side and all terms involving \(x\) are on the other side. This means rewriting the equation as \(y \, dy = \sec(y^{2}) \sec^{2}(x) \, dx\) and then rearranging to isolate \(y\) and \(dy\) on one side and \(x\) and \(dx\) on the other.
3. Look carefully at the terms to see if you can write the equation in the form \(f(y) \, dy = g(x) \, dx\). Since \(\sec(y^{2})\) depends on \(y\), try to express the left side as a function of \(y\) only and the right side as a function of \(x\) only.
4. Integrate both sides separately: \(\int f(y) \, dy = \int g(x) \, dx\). This will involve integrating \(y\) and \(\sec(y^{2})\) with respect to \(y\), and \(\sec^{2}(x)\) with respect to \(x\).
5. After integration, include the constant of integration \(C\) and write the implicit solution relating \(y\) and \(x\). If possible, solve for \(y\) explicitly, but often the solution may remain implicit.

Key Concepts

Separable Differential Equations

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

Integration of Trigonometric Functions

Solving the given equation involves integrating functions like secant and secant squared. Knowing the antiderivatives of these functions, such as ∫sec²(x) dx = tan(x), is essential for correctly integrating both sides after separation.

Implicit Solutions of Differential Equations

Sometimes, after integration, the solution cannot be explicitly solved for y. Understanding implicit solutions means recognizing and working with equations where y is defined implicitly in terms of x, which is common in nonlinear differential equations.