Differential Equations - am I doing this right

[protonchain]

Homework Statement

What is y(x) [y as a function of x] given that

Homework Equations

\frac{dy}{dx} = 3\frac{y}{x}

where you are given the boundary condition that y = y_0 at x = x_0 where y_0 and x_0 are constants.

The Attempt at a Solution

Separating variables \Rightarrow \frac{dy}{y} = \frac{3 dx}{x}

Integrating \Rightarrow \int{\frac{dy}{y}} = \int{\frac{3 dx}{x}

Integrals give \Rightarrow ln(y) = 3 ln(x) + C

Given the boundary conditions then \Rightarrow ln(y_0) = 3 ln(x_0) + C

Therefore \Rightarrow C = ln(y_0) - 3ln(x_0) = ln(\frac{y_0}{x_0^3})

Plugging back into ln(y) = 3 ln(x) + C gives
ln(y) = 3 ln(x) + ln(\frac{y_0}{x_0^3})

Therefore \Rightarrow y = exp\left(ln(x^3) + ln(\frac{y_0}{x_0^3})\right)

\Rightarrow y = exp\left(ln(x^3 \bullet \frac{y_0}{x_0^3})\right) \Rightarrow y = y_0(\frac{x}{x_0})^3

 

[jamesrc]

Looks good to me.

One way you can check your work is to find dy/dx from your final equation, and verify that you get your initial equation...

 

[protonchain]

That's the part where I got worried about whether I did it right or not.

I think I was getting stucking trying to prove that it was correct, and I just did it.

Thank you for the assurance and the help! :) Really appreciate it