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protonchain
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Homework Statement
What is [tex] y(x) [/tex] [[tex] y [/tex] as a function of [tex] x [/tex]] given that
Homework Equations
[tex] \frac{dy}{dx} = 3\frac{y}{x} [/tex]
where you are given the boundary condition that [tex] y = y_0 [/tex] at [tex] x = x_0 [/tex] where [tex] y_0 [/tex] and [tex] x_0 [/tex] are constants.
The Attempt at a Solution
Separating variables [tex] \Rightarrow \frac{dy}{y} = \frac{3 dx}{x} [/tex]
Integrating [tex] \Rightarrow \int{\frac{dy}{y}} = \int{\frac{3 dx}{x}[/tex]
Integrals give [tex] \Rightarrow ln(y) = 3 ln(x) + C [/tex]
Given the boundary conditions then [tex] \Rightarrow ln(y_0) = 3 ln(x_0) + C [/tex]
Therefore [tex] \Rightarrow C = ln(y_0) - 3ln(x_0) = ln(\frac{y_0}{x_0^3}) [/tex]
Plugging back into [tex] ln(y) = 3 ln(x) + C [/tex] gives
[tex] ln(y) = 3 ln(x) + ln(\frac{y_0}{x_0^3}) [/tex]
Therefore [tex] \Rightarrow y = exp\left(ln(x^3) + ln(\frac{y_0}{x_0^3})\right) [/tex]
[tex] \Rightarrow y = exp\left(ln(x^3 \bullet \frac{y_0}{x_0^3})\right) \Rightarrow y = y_0(\frac{x}{x_0})^3[/tex]