Linear First Order ODE, Bernoulli Equations and Applications

Linear First Order ODE
DE of the form: y'+P(x)y=Q(x).

Integrating factor: R(x)=e^{\int P(x)\,dx}.
\begin{aligned}  R'&=RP\\  Ry'+RPy&=RQ\\  (Ry)'&=RQ\\  Ry&=\int RQ\,dx  \end{aligned}
\displaystyle \boxed{y=\frac{\int RQ\,dx}{R}}
(Remember to have a constant C when integrating the numerator \int RQ\,dx.)

Integration by parts
\displaystyle  \boxed{\int uv'\,dx=uv-\int u'v\,dx}

Acronym: LIATE (Log, Inverse Trig., Algebraic, Trig., Exponential), where L is the best choice for u. (This is only a rough guideline.)

Bernoulli Equations
DE of the form: y'+p(x)y=q(x)y^n.

y^{-n}y'+y^{1-n}p(x)=q(x)

Set \boxed{y^{1-n}=z}.

Then (1-n)y^{-n}y'=z'. The given DE becomes
\displaystyle  \boxed{z'+(1-n)p(x)z=(1-n)q(x)}.

Fundamental Theorem of Calculus (FTC)
Part 1: \displaystyle \frac{d}{dx}\int_a^x f(t)\,dt=f(x)

Part 2: \displaystyle \int_a^b F'(t)\,dt=F(b)-F(a)

Hyperbolic Functions
\begin{aligned}  \sinh x&=\frac{e^x-e^{-x}}{2}\\  \cosh x&=\frac{e^x+e^{-x}}{2}\\  \cosh^2 x-\sinh^2 x&=1\\  \end{aligned}
\begin{aligned}  \frac{d}{dx}\sinh x&=\cosh x\\  \frac{d}{dx}\cosh x&=\sinh x\\  \frac{d}{dx}\sinh^{-1}x&=\frac{1}{\sqrt{x^2+1}}\\  \frac{d}{dx}\cosh^{-1}x&=\frac{1}{\sqrt{x^2-1}}  \end{aligned}
\displaystyle \int \tanh(ax)\,dx=\frac{1}{a}\ln(\cosh(ax))+C.

Uranium-Thorium Dating
Starting Equations:
\displaystyle \begin{cases}  \frac{dU}{dt}=-k_U U\implies U=U_0e^{-k_Ut}\\  \frac{dT}{dt}=k_UU-k_TT.  \end{cases}

\frac{dT}{dt}+k_T T=k_U U_0e^{-k_Ut}

R=e^{\int k_T}=e^{k_T t}.

\displaystyle \boxed{T(t)=\frac{k_U}{k_T-k_U}U_0(e^{-k_Ut}-e^{-k_Tt})}

\displaystyle \boxed{\frac{T}{U}=\frac{k_U}{k_T-k_U}[1-e^{(k_U-k_T)t}]}

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