- #1

fluidistic

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## Homework Statement

1)Find the solution of [tex]x'=x^{\frac{1}{2}}[/tex] that passes through the point [tex](t_0, x_0)[/tex] where [tex]x_0>0[/tex].

2)Find all the solutions of this equation that pass through the point [tex](t_0,0)[/tex].

## Homework Equations

Direct integration.

## The Attempt at a Solution

1)[tex]\frac{dx}{dt}=x^{\frac{1}{2}} \Rightarrow \frac{dx}{x^{\frac{1}{2}}}=dt \Rightarrow \int \frac{dx}{x^{\frac{1}{2}}}=t+C\Rightarrow 2 x^{\frac{1}{2}}=t+C \Rightarrow x=\frac{t^2}{4}+tC+C^2[/tex].

I determined C thanks to the initial condition and the equation became [tex]x=\frac {t^2}{4}+t \left ( \frac{2x_0 ^{\frac{1}{2}}-t_0}{2}} \right ) + \frac{(2x_0 ^{\frac{1}{2}}-t_0)^2}{4}}[/tex].

2) Replacing [tex]x_0[/tex] by [tex]0[/tex] in the above equation yields [tex]x= \left ( \frac{t}{2}-\frac{t_0}{2} \right ) ^2[/tex].

Unfortunately I replaced this solution into the original equation and the equality isn't true. So I made an error. I also replaced the first solution I got (the one with C's) into the equation and it didn't work. Hence I made an error quite early. I don't know where though.