Differential equation

Question

1) Consider x ≠ 0. Solve the differential equation

(y⁶ − 4x⁶)dx + 3x⁴y²dy = 0.

Determine the particular solution that passes through the point (1, 2).

Answer 1

1) (y⁶ − 4x⁶)dx + 3x⁴y²dy = 0, dy/dx = (4x⁶ - y⁶)/(3x⁴y²). This is homogenous equation: y = vx, v is new function, dy/dx = v + xdv/dx. We substitute in given ODE>

v + xdv/dx = (4x⁶ - x⁶v⁶)/(3x⁴·x²v²), or xdv/dx = (4 - v⁶)/(3v²) - v; xdv/dx = (4 - 3v³ - v⁶)/(3v²);

3v²dv/(4 - 3v³ - v⁶) = dx/x; 3v²dv/(v³ + 4)(v³ - 1) = - dx/x; - ∫3v²dv/5·(1/(v³ + 4) - 1/(v³ - 1)) = - ∫dx/x;

-1/5n(v³ + 4) + 1/5ln(v³ - 1) = - lnx - 1/5lnC; (v³ + 4)/(v³ - 1) = Cx⁵;

(y³/x³ + 4)(y³/x³ - 1) = Cx⁵.

(1,2) IVP: (8 + 4)(8 - 1) = C; C = 84.

(y³/(x³ + 4)(y³/x³ - 1) = 84x⁵.