Show that cx^3 is a family of solutions ( differential equations)

Question

Consider the differential equation xy'-3y=0 

 

1) Show that cx^3 is a family of solutions. 

2) Find the particular solution that satisfies y(-3)=2 

Answer 1

xy' - 3y = 0

 

  y' - (3/x)y = 0 ← linear first order differential equation

 

The integrating factor is e^∫(-3/x)dx = e^-3lnx

 

                                               = e^{ln(x^-3)}

 

                                               = x^-3

 

Multiply the differential equation by x^-3 to obtain:

 

x^-3y' - 3x^-4y = 0

 

So, (d/dx)(x^-3y) = 0

 

              x^-3y = C

 

                 y = Cx³     y(x) = Cx³

 

                                    If y(-3) = 2, then C(-3)³ = 2

 

                                                             C = -2/27

 

                                                  y(x) = (-2/27)x³