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

The domain of the particular solution to the differential equation is (-∞, 0) U (0, ∞).

Step-by-step explanation:

The given differential equation is dy/dx = (3y/x). To find the domain of the particular solution, we need to consider the values of x and y that satisfy the equation.

Since the denominator x cannot be zero, the domain of the solution is all real numbers except x=0. Therefore, the domain is (-∞, 0) U (0, ∞).

Answer 2

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