Solve to get explicit form

Question

Use an appropriate method to solve the following 1st order ODE:

  

Express the final answer in an explicit form. Please show ALL work to get credits.

Answer 1

This is Homojinius equation. Substitution y= xv, where v is new function of x. Then dy/dx = y' = v + xv'/

We put this in our equation: v + xv' = (3x² + 2x²v²)/(x²v); v + xv' = (3 + 2v²)/v; xv' = 3/v + v;

xdv/dx = (3 + v²)/v; Now we can separate variables: v/(v² + 3)dv = 1/xdx.

We takes integrals now: ∫v/(v² + 3)dv = ∫1/xdx; 1/2ln(v² + 3) = ln(absx) + 1/2lnC; ln(v² + 3) = 2ln(absx) + LnC;

v² + 3 = Cx², v² = Cx² - 3; y²/x² = Cx² - 3, y² = Cx⁴ - 3x² and y = ± (Cx⁴ - 3x²)^1/2.

Answer: general solution of this ODE: y = ± (Cx⁴ - 3x²)^1/2.