Question

Find the domain for the particular solution to the differential equation dy dx equals the quotient of 3 times y and x , with initial condition y(1) = 1.

Answer 1

For this case we have the following difference equation:
 $dy / dx = 3xy$
 Applying separable variables we have:
 $dy / y = 3xdx$
 Integrating both sides we have:
 $\int\ ({1/y}) \, dy = \int\ {3x} \, dx$
 $ln (y) = (3/2) x ^ 2 + C$
 applying exponential to both sides:
 $exp (ln (y)) = exp ((3/2) x ^ 2 + C) y = C * exp ((3/2) x ^ 2)$
 For y (1) = 1 we have:
 $C = 1 / (exp ((3/2) * 1 ^ 2)) C = 0.2$
 Thus, the particular solution is:
 $y = 0.2 * exp ((3/2) x ^ 2)$
 Whose domain is all real.
 Answer:
 y = 0.2 * exp ((3/2) x ^ 2)
 Domain: all real numbers