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

Elementary Differential Equations 2.3 Problem 1: Separation of Variables

Find the complete solution of the differential equation $latex \frac{dy}{dx}=\frac{2x+2xy^2}{y+2x^2y}&s=1&fg=000000$. SOLUTION: $latex \frac{dy}{dx}=\frac{2x+2xy^2}{y+2x^2y}&s=3&fg=000000$ By cross-multiplication, $latex \left(y+2x^2y\right)dy=\left(2x+2xy^2\right)dx&s=0&fg=000000$ Factor out common monomial factors $latex y\left(1+2x^2\right)dy=2x\left(1+y^2\right)dx&s=0&fg=000000$ Simplify $latex 2x\left(1+y^2\right)dx-y\left(1+2x^2\right)dy=0&s=0&fg=000000$ Combine all terms with x, and combine all terms with y $latex \frac{2xdx}{\left(1+2x^2\right)}-\frac{ydy}{\left(1+y^2\right)}=0&s=2&fg=000000$ Since the variables are already separated, we can already apply integration. That is, $latex \int \frac{2dx}{1+2x^2\:}-\int … Continue reading Elementary Differential Equations 2.3 Problem 1: Separation of Variables