Exercise
$\frac{dy}{dx}=1.\frac{2y-x}{y+2x}$
Step-by-step Solution
Learn how to solve implicit differentiation problems step by step online. Solve the differential equation dy/dx=1(2y-x)/(y+2x). Multiplying the fraction by 1. We can identify that the differential equation \frac{dy}{dx}=\frac{2y-x}{y+2x} 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.
Solve the differential equation dy/dx=1(2y-x)/(y+2x)
Final answer to the exercise
$-\frac{1}{2}\ln\left(1+\left(\frac{y}{x}\right)^2\right)-2\arctan\left(\frac{y}{x}\right)=\ln\left(x\right)+C_0$