Notes: `(ds)/(dalpha) = sin^2(alpha/2)cos^2(alpha/2)` Solve the differential equation.

Tuesday, 21 July 2015

$\displaystyle\frac{{{d}{s}}}{{{d}\alpha}}={{\sin}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}{{\cos}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}$ Solve the differential equation.

$\displaystyle\frac{{{d}{s}}}{{{d}\alpha}}={{\sin}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}{{\cos}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}$

To solve, express the differential equation in the form N(y)dy = M(x)dx .

So bringing together same variables on one side, the equation becomes

$\displaystyle{d}{s}={{\sin}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}{{\cos}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}{d}\alpha$

To simplify the right side, apply the exponent rule $\displaystyle{{\left({a}{b}\right)}}^{{m}}={{a}}^{{m}}{{b}}^{{m}}$ .

$\displaystyle{d}{s}={{\left({\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}\right)}}^{{2}}{d}\alpha$

Then, apply the sine double angle identity $\displaystyle{\sin{{\left({2}\theta\right)}}}={2}{\sin{{\left(\theta\right)}}}{\cos{{\left(\theta\right)}}}$ .

$\displaystyle{\sin{{\left({2}\cdot\frac{\alpha}{{2}}\right)}}}={2}{\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}$

$\displaystyle{\sin{{\left(\alpha\right)}}}={2}{\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}$

$\displaystyle{\sin{{\left(\alpha\right)}}}={2}{\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}$

$\displaystyle\frac{{{\sin{{\left(\alpha\right)}}}}}{{2}}={\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}$

Substituting this to the right side, the differential equation becomes

$\displaystyle{d}{s}={{\left(\frac{{{\sin{{\left(\alpha\right)}}}}}{{2}}\right)}}^{{2}}\ldots$

$\displaystyle\frac{{{d}{s}}}{{{d}\alpha}}={{\sin}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}{{\cos}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}$

To solve, express the differential equation in the form N(y)dy = M(x)dx .

So bringing together same variables on one side, the equation becomes

$\displaystyle{d}{s}={{\sin}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}{{\cos}}^{{2}}{\left(\frac{\alpha}{{2}}\right)}{d}\alpha$

To simplify the right side, apply the exponent rule $\displaystyle{{\left({a}{b}\right)}}^{{m}}={{a}}^{{m}}{{b}}^{{m}}$ .

$\displaystyle{d}{s}={{\left({\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}\right)}}^{{2}}{d}\alpha$

Then, apply the sine double angle identity $\displaystyle{\sin{{\left({2}\theta\right)}}}={2}{\sin{{\left(\theta\right)}}}{\cos{{\left(\theta\right)}}}$ .

$\displaystyle{\sin{{\left({2}\cdot\frac{\alpha}{{2}}\right)}}}={2}{\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}$

$\displaystyle{\sin{{\left(\alpha\right)}}}={2}{\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}$

$\displaystyle{\sin{{\left(\alpha\right)}}}={2}{\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}$

$\displaystyle\frac{{{\sin{{\left(\alpha\right)}}}}}{{2}}={\sin{{\left(\frac{\alpha}{{2}}\right)}}}{\cos{{\left(\frac{\alpha}{{2}}\right)}}}$

Substituting this to the right side, the differential equation becomes

$\displaystyle{d}{s}={{\left(\frac{{{\sin{{\left(\alpha\right)}}}}}{{2}}\right)}}^{{2}}{d}\alpha$

$\displaystyle{d}{s}=\frac{{{{\sin}}^{{2}}{\left(\alpha\right)}}}{{4}}{d}\alpha$

Then, apply the cosine double angle identity $\displaystyle{\cos{{\left({2}\theta\right)}}}={1}-{2}{{\sin}}^{{2}}{\left(\theta\right)}$ .

$\displaystyle{\cos{{\left({2}\alpha\right)}}}={1}-{2}{{\sin}}^{{2}}{\left(\alpha\right)}$

$\displaystyle{2}{{\sin}}^{{2}}{\left(\alpha\right)}={1}-{\cos{{\left({2}\alpha\right)}}}$

$\displaystyle{{\sin}}^{{2}}{\left(\alpha\right)}=\frac{{{1}-{\cos{{\left({2}\alpha\right)}}}}}{{2}}$

Plugging this to the right side, the differential equation becomes

$\displaystyle{d}{s}=\frac{{\frac{{{1}-{\cos{{\left({2}\alpha\right)}}}}}{{2}}}}{{4}}{d}\alpha$

$\displaystyle{d}{s}=\frac{{{1}-{\cos{{\left({2}\alpha\right)}}}}}{{8}}{d}\alpha$

$\displaystyle{d}{s}={\left(\frac{{1}}{{8}}-\frac{{\cos{{\left({2}\alpha\right)}}}}{{8}}\right)}{d}\alpha$

Then, take the integral of both sides.

$\displaystyle\int{d}{s}=\int{\left(\frac{{1}}{{8}}-\frac{{\cos{{\left({2}\alpha\right)}}}}{{8}}\right)}{d}\alpha$

$\displaystyle\int{d}{s}=\int\frac{{1}}{{8}}{d}\alpha-\int\frac{{\cos{{\left({2}\alpha\right)}}}}{{8}}{d}\alpha$

Apply the integral formulas $\displaystyle\int{a}{\left.{d}{x}\right.}={a}{x}+{C}$ and $\displaystyle\int{\cos{{\left({x}\right)}}}{\left.{d}{x}\right.}={\sin{{\left({x}\right)}}}+{C}$ .

$\displaystyle{s}+{C}_{{1}}=\frac{{1}}{{8}}\alpha-\frac{{{\sin{{\left({2}\alpha\right)}}}}}{{16}}+{C}_{{2}}$

Then, isolate the s.

$\displaystyle{s}=\frac{{1}}{{8}}\alpha-\frac{{{\sin{{\left({2}\alpha\right)}}}}}{{16}}+{C}_{{2}}-{C}_{{1}}$

Since C1 and C2 represents any number, it can be expressed as a single constant C.

$\displaystyle{s}=\frac{{1}}{{8}}\alpha-\frac{{{\sin{{\left({2}\alpha\right)}}}}}{{16}}+{C}$

Therefore, the general solution of the differential equation is $\displaystyle{s}=\frac{{1}}{{8}}\alpha-\frac{{{\sin{{\left({2}\alpha\right)}}}}}{{16}}+{C}$ .