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