## 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