# Thread: differential equation w/ initial condition

1. ## differential equation w/ initial condition

Solve the differential equation with given conditions

(dy/dt)=(1/2)y

y(2)=100

2. Originally Posted by thedoge
Solve the differential equation with given conditions

(dy/dt)=(1/2)y

y(2)=100
$\displaystyle y'=\frac{1}{2}y$
Divide by $\displaystyle y$.
First check if $\displaystyle y=0$ is a solution (it is).
Now look for other solutions $\displaystyle y\not = 0$,
$\displaystyle \frac{y'}{y}=\frac{1}{2}$
Integrate,
$\displaystyle \int \frac{y'}{y} dx = \int\frac{1}{2} dx$
Thus,
$\displaystyle \ln |y|=\frac{1}{2}x+C'$
Thus,$\displaystyle e^{\ln |y|}=e^{\frac{1}{2}x+C'}$
$\displaystyle |y|=Ce^{\frac{1}{2}x},C>0$
$\displaystyle y=Ce^{\frac{1}{2}x}$
$\displaystyle 100=Ce^{\frac{1}{2}(2)}$
$\displaystyle 100=Ce^{1}=Ce$
$\displaystyle C=100/e=100e^{-1}$.
Thus,
$\displaystyle y=Ce^{\frac{1}{2}x}=100e^{-1}e^{\frac{1}{2}x}=100e^{\frac{1}{2}x-1}$

3. As fast and amazing as usual PH.

If you don't mind me asking, do you have a specific profession outside this forum? You seem to be a master of mathematics which should open up about any occupation to you

4. Originally Posted by thedoge

If you don't mind me asking, do you have a specific profession outside this forum?
Yes! I am a fashion designer.

5. Haha. A comedian too.

Unless you're serious. One can never know online;]

6. Hrm. I follow everything you said perfectly and even worked it out that way myself, but apparently this problem needs to evaluate to a number.

What is the value of x?

Nevermind. I see what the problem was.

*** for future reader's reference replace the x in 100*e^(.5x-1) with a t