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We can identify that the differential equation $\frac{dy}{dx}=\frac{-\left(4x+3y\right)}{2x+y}$ is homogeneous, since it is written in the standard form $\frac{dy}{dx}=\frac{M(x,y)}{N(x,y)}$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and both are homogeneous functions of the same degree
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$\frac{dy}{dx}=\frac{-\left(4x+3y\right)}{2x+y}$
Learn how to solve problems step by step online. dy/dx=(-(4x+3y))/(2x+y). We can identify that the differential equation \frac{dy}{dx}=\frac{-\left(4x+3y\right)}{2x+y} is homogeneous, since it is written in the standard form \frac{dy}{dx}=\frac{M(x,y)}{N(x,y)}, where M(x,y) and N(x,y) are the partial derivatives of a two-variable function f(x,y) and both are homogeneous functions of the same degree. Use the substitution: y=ux. Expand and simplify. Apply the formula: b\cdot dy=a\cdot dx\to \int bdy=\int adx, where a=\frac{1}{x}, b=\frac{2+u}{-\left(u+1\right)\left(u+4\right)}, dy=du, dyb=dxa=\frac{2+u}{-\left(u+1\right)\left(u+4\right)}du=\frac{1}{x}dx, dyb=\frac{2+u}{-\left(u+1\right)\left(u+4\right)}du and dxa=\frac{1}{x}dx.
