Exercise
$\frac{dy}{dx}=\frac{-2x^3y}{x^4+y^4}$
Step-by-step Solution
Learn how to solve simplify trigonometric expressions problems step by step online. Solve the differential equation dy/dx=(-2x^3y)/(x^4+y^4). We can identify that the differential equation \frac{dy}{dx}=\frac{-2x^3y}{x^4+y^4} 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. 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 dy/dx=(-2x^3y)/(x^4+y^4)
Final answer to the exercise
$-\frac{1}{6}\ln\left|\frac{3x^{4}}{y^{4}}+1\right|=\ln\left|y\right|+C_0$