Separation of Variables| Elementary Differential Equations|dela Fuente, Feliciano, and Uy|Example 2

Find the complete solution of the following differential equation:

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Separation of Variables| Elementary Differential Equations|dela Fuente, Feliciano, and Uy|Problem 1

Find the solution of the following differential equation:  $latex \frac{dy}{dx}=\frac{4x+xy^2}{y-x^2y}&s=2&fg=000000$ SOLUTION: $latex \frac{dy}{dx}=\frac{4x+xy^2}{y-x^2y}&s=2&fg=000000$ $latex \left(y-x^2y\right)dy=\left(4x+xy^2\right)dx&s=1&fg=000000$ $latex y\left(1-x^2\right)dy=x\left(4+y^2\right)dx&s=1&fg=000000$ $latex \frac{ydy}{4+y^2\:}=\frac{xdx}{1-x^2}&s=2&fg=000000$ $latex \frac{ydy}{4+y^2\:}-\frac{xdx}{1-x^2}=0&s=2&fg=000000$ $latex \int \frac{ydy}{4+y^2\:}-\int \frac{xdx}{1-x^2}=\int 0&s=2&fg=000000$ There are two integrals, and we will perform them separately. For $latex \int \frac{ydy}{4+y^2\:}&s=2&fg=000000$ Let $latex u=4+y^2&s=1&fg=000000$, so $latex du=2ydy&s=1&fg=000000$ $latex \int \:\frac{ydy}{4+y^2}=\int \:\frac{\frac{1}{2}du}{u}=\frac{1}{2}\int \:\frac{du}{u}&s=2&fg=000000$ $latex =\frac{1}{2}ln\left(u\right)+C_1&s=1&fg=000000$ $latex =\frac{1}{2}ln\left(4+y^2\right)+C_1&s=1&fg=000000$ For $latex … Continue reading Separation of Variables| Elementary Differential Equations|dela Fuente, Feliciano, and Uy|Problem 1