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Solve dx/dt = = (1+sqrt(t))/(1+sqrt(x))

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\[\frac{ dx }{ dt } = \frac{ 1+\sqrt{t} }{ 1+\sqrt{x} }\]
i ended up with something, but I cant explicitly solve for t
The way we were tough is to separate the variables and then integrate each side.

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that's not how we were taught to do it..
hmm sorry i have not yet learned those so sorry what chapter is this
What chapter? It would depend what textbook. And what class.
i mean name of the chapter
its on ODE's
is this college calculus
yeah calc 2
It's separable, no?
:O I think it is separable!
Multiply both sides by \(1+\sqrt{x}\). Integrate with respect to \(t\).
This is very clearly a separable first-order ordinary differential equation. We can easily separate as follows: $$\frac{dx}{dt}=\frac{1+\sqrt{t}}{1+\sqrt{x}}\\(1+\sqrt{x})\ dx=(1+\sqrt{t})\ dt$$Now, it should be clear that we integrate both sides.$$\int(1+\sqrt{x})\ dx=\int(1+\sqrt{t})\ dt\\x+\frac23x^\frac32=t+\frac23t^\frac32+C$$ This yields an implicit solution; for an explicit one, you'll need to isolate \(x\)... it won't be pretty.
@oldrin.bataku thanks for the reply, that is the same thing I did, but I couldn't figure out how to get an explicit solution. I suppose I will leave it at this. Thank you
I *highly* doubt your teacher wants an explicit solution ;-)

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