Autonomous Differential Equation

n.a.s.h
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Homework Statement


solve the following...


dy/dt= 0.5(100-y) with initial condition:y(0)=20

Homework Equations





The Attempt at a Solution



i found the general equation to be I=0.5Ln |100-y|+c
but I am stuck as to what to do with the initial condition...
 
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n.a.s.h said:

Homework Statement


solve the following...


dy/dt= 0.5(100-y) with initial condition:y(0)=20

Homework Equations





The Attempt at a Solution



i found the general equation to be I=0.5Ln |100-y|+c
but I am stuck as to what to do with the initial condition...

What's "I"?

And you are saying that 0.5 * ln |100-y| + c is some kind of a solution to this DE?

\frac{dy}{dt} = \frac{100-y}{2}
 
yes i think its the general equation...
 
n.a.s.h said:
yes i think its the general equation...

I would say put t=0 and y=20 into that solution and then solve for the constant c, but I can't. Because there is no t. That's a problem. Can you show us how you got it? I think there's some other errors on the way.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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