Trigonometric Identities and Differential Equations

Trigonometric Identities
$\begin{aligned} \sin(A\pm B)&=\sin A\cos B\pm\cos A\sin B\\ \cos(A\pm B)&=\cos A\cos B\mp\sin A\sin B\\ \tan(A\pm B)&=\frac{\tan A\pm \tan B}{1\mp \tan A\tan B} \end{aligned}$
$\begin{aligned} \sin A+\sin B&=2\sin\frac{1}{2}(A+B)\cos\frac{1}{2}(A-B)\\ \sin A-\sin B&=2\cos\frac{1}{2}(A+B)\sin\frac{1}{2}(A-B)\\ \cos A+\cos B&=2\cos\frac{1}{2}(A+B)\cos\frac{1}{2}(A-B)\\ \cos A-\cos B&=-2\sin\frac{1}{2}(A+B)\sin\frac{1}{2}(A-B) \end{aligned}$

Reduction to Separable Form
Equations of the form
$y'=g(\frac{y}{x}).$
Set $\frac{y}{x}=v$ . Then $y=vx$ and $y'=v+xv'$ . Thus the original equation becomes $v+xv'=g(v)$ which is separable.

Linear Change of Variable
$y'=f(ax+by+c)$ can be solved by setting $u=ax+by+c$ .

Newton’s Law of Cooling
Rate of change of temperature of an object ( $\frac{dT}{dt}$ ) is proportional to the difference between its own temperature ( $T$ ) and the ambient temperature ( $T_\text{env}$ ). That is, $\frac{dT}{dt}=-k(T-T_\text{env})$ .

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