Question

4. Solve the differential equation 4xy dx/dy=y2−1

Answer 1

`\displaystyle x=(\pm√(y^2-\ln(y^2)+C))/(2)`

Explanation:

`\displaystyle 4xy(dx)/(dy)=y^2-1\\\\4x(dx)/(dy)=y-(1)/(y)\\\\4x\,dx=\biggr(y-(1)/(y)\biggr)\,dy\\\\\int4x\,dx=\int\biggr(y-(1)/(y)\biggr)\,dy\\\\2x^2=(y^2)/(2)-\ln(|y|)+C\\\\4x^2=y^2-2\ln(|y|)+C\\\\4x^2=y^2-\ln(y^2)+C\\\\x^2=(y^2-\ln(y^2)+C)/(4)\\\\x=(\pm√(y^2-\ln(y^2)+C))/(2)`