Calculus help please

Question

Find the domain for the differential equation dy/dx = 3y/x with the initial condition y(1)=1

Answer 1

Hi Bob,

dy/dx = 3y/x ⇒ (1/y)dy = (3/x)dx ⇒ ∫ (1/y)dy = ∫ (3/x)dx ⇒ ln |y| = 3ln|x| + C ⇒ |y| = C|x³| ⇒ y = ±Cx³

Initial condition; y(1) = 1 ⇒ 1 = C*1 ⇒ C = 1.

Thus your DE is y = ± x³. Which is defined on (-∞, ∞), but x = 0 because you can't divide with a 0.

Therefore, the answer is (-∞, 0) ∪ (0, ∞).