Exercise
$\left(2x\:+y\right)\cdot dx-\left(x+6y\right)\cdot dy=0$
Step-by-step Solution
Learn how to solve problems step by step online. Solve the differential equation (2x+y)dx-(x+6y)dy=0. We can identify that the differential equation \left(2x+y\right)dx-\left(x+6y\right)dy=0 is homogeneous, since it is written in the standard form M(x,y)dx+N(x,y)dy=0, 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: x=uy. Expand and simplify. Integrate both sides of the differential equation, the left side with respect to u, and the right side with respect to y.
Solve the differential equation (2x+y)dx-(x+6y)dy=0
Final answer to the exercise
$-2\ln\left|\frac{\sqrt{-x^2+3y^2}}{\sqrt{3}y}\right|+\frac{\sqrt{3}\ln\left|\frac{\sqrt{3}\left(\frac{x}{\sqrt{3}y}+1\right)y}{x-\sqrt{3}y}\right|}{6}=2\ln\left|y\right|+C_0$