solving differential equations under the given conditions?

Find the general solution by separating the variables then integrating:

dy / dx = 3x√(1 + y²) / y

y / √(1 + y²) dy = 3x dx

∫ 2y / √(1 + y²) dy / 2 = 3 ∫ x dx

√(1 + y²) = 3x² / 2 + C

1 + y² = (3x² / 2 + C)²

1 + y² = [(3x² + C) / 2]²

1 + y² = (3x² + C)² / 4

y² = (3x² + C)² / 4 - 1

y = ±√[(3x² + C)² / 4 - 1]

y = ±√[{(3x² + C)² - 4} / 4]

y = ±√[(3x² + C)² - 4] / 2

y = ±√[{3(x² + C)}² - 4] / 2

y = ±√[9(x² + C)² - 4] / 2

Find the particular solution by solving for the constant:

When x = 1, y = √8

±√[9(1 + C)² - 4] / 2 = √8

±√[9(1 + C)² - 4] = 2√8

9(1 + C)² - 4 = 32

9(1 + C)² = 36

(1 + C)² = 4

1 + C = ±2

C = -3 OR C = 1

Only the positive square root works hence,

y = {√[9(x² - 3)² - 4] / 2, √[9(x² + 1)² - 4] / 2}

dy/dx = 3x√(y² + 1)/y

y dy/√(y² + 1) = 3x dx

Integrating both sides:

√(y² + 1) = 3x²/2 + C

y = +/-√[(3x²/2 + C)² - 1]