Question

Find the domain for the particular solution to the differential dy/dx = 3y/x, with initial condition

x>0
x<0
|x|<=0
x is not equal to 0
all real numbers

Answer 1

the equation is \[\frac{ dy }{ dx }=\frac{ 3y }{ x }\]

Answer 2

\(\frac{dy}{dx}=\frac{3y}{x}\implies\int\frac{1}{3y}dy=\int\frac{1}{x}dx\implies\frac{\ln(y)}{3}=\ln(x)+c\\\implies ln(y)=3\ln(x)+c_0\implies y = e^{3ln(x)+c_0}\implies y = e^{3\ln(x)}e^{c_0}\\\implies y = c_1e^{\ln(x^3)}\implies y=c_1x^3\)

Answer 3

domain range is all real numbers for \(c_1\ne 0\)