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$\frac{dy}{dx} = 3\sin(x/2)$

$y = 1 ,x= \frac{\pi}{3}$

I just get stuck on the integration of the $3\sin(x/2)$.

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    okay but what is the integration of 3sin(x/2)?2012-07-24

3 Answers 3

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$\int\sin kx\,dx=-\frac{1}{k}\cos kx+C\,\,,\,k\neq 0\,$

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Begin by getting the general solution, $y = \int 3\sin(x/2)\, dx $ You can then use the second condition to solve for the constant of integration and obtain a solution to the DE.

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    @Kudla69, what *exactly* isn't clear to you in my answer above? I'm giving you there the answer! If you're beginning with integrals it might be that you're not that ready to go on differential equations: for these you need high proficiency in integration.2012-07-24
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Here $\displaystyle \frac{dy}{dx}=3\sin \frac{x}{2}$.

$\displaystyle \Rightarrow dy= 3\sin \frac{x}{2} dx$ (variable saperable)

$\displaystyle \Rightarrow \int dy=3\int \sin \frac{x}{2} dx+ C$

$\displaystyle \therefore y=-3 \frac{\cos{\frac{x}{2}}}{\frac{1}{2}}+C$.

i.e. $\displaystyle y=-6\cos \frac{x}{2}+C$.

Now, $\displaystyle y=1$ and $\displaystyle x=\frac{\pi}{3}$, we get

$\displaystyle 1=-6\cos \frac{\pi}{6} +C$ $\Rightarrow 1=-6 \frac{\sqrt{3}}{2}+C$ $\Rightarrow C=1+3\sqrt{3}.$

The particular solution of give differential equation is $\displaystyle y=-6\cos \frac{x}{2}+1+3\sqrt{3}$.