Question

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

Accepted Answer

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}$$(x). $$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. 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. 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 $$x. $$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.

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

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Key Concepts

Separable Differential Equations

Integration of Trigonometric Functions

Implicit Solutions of Differential Equations