Exercise
$\frac{dy}{dx}\cdot\left(y-x\right)=x+y$
Step-by-step Solution
Learn how to solve problems step by step online. Solve the differential equation dy/dx(y-x)=x+y. Rewrite the differential equation. We can identify that the differential equation \frac{dy}{dx}=\frac{x+y}{y-x} 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: x=uy. Expand and simplify.
Solve the differential equation dy/dx(y-x)=x+y
Final answer to the exercise
$\ln\left|\frac{\sqrt{2}y}{\sqrt{\left(x+y\right)^2-2y^2}}\right|=\ln\left|y\right|+C_0$