# 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: $\displaystyle \begin{cases} \frac{dU}{dt}=-k_U U\implies U=U_0e^{-k_Ut}\\ \frac{dT}{dt}=k_UU-k_TT. \end{cases}$ $\frac{dT}{dt}+k_T T=k_U U_0e^{-k_Ut}$ $R=e^{\int k_T}=e^{k_T t}$. $\displaystyle \boxed{T(t)=\frac{k_U}{k_T-k_U}U_0(e^{-k_Ut}-e^{-k_Tt})}$ $\displaystyle \boxed{\frac{T}{U}=\frac{k_U}{k_T-k_U}[1-e^{(k_U-k_T)t}]}$ ## Author: mathtuition88

https://mathtuition88.com/

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