<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/be3d8d2a-03fc-444d-b176-a2e0a77be13b/Differential_equations.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/be3d8d2a-03fc-444d-b176-a2e0a77be13b/Differential_equations.png" width="40px" /> Differential Equations: are equations where there is both the function and it’s derivative, and you solve for the function
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$$ \frac{\text{d}y}{\text{d}x}=f(x)g(y) $$
$$ \begin{aligned} \frac{\text{d}y}{\text{d}x}&=f(x)g(y) \\ \frac{\text{d}y}{g(y)}&=f(x)\,\text{d}x \\ \int\frac{1}{g(y)}\,\text{d}y&=\int f(x)\,\text{d}x \end{aligned} $$
Example:
$$ \begin{aligned} \frac{\text{d}N}{\text{d}t}&=-\lambda N \qquad [\text{find }N(t)] \\ \int\frac{1}{N}\,\text{d}N&=-\int\lambda\,\text{d}t \\ \ln(N)&=-\lambda t+c \\ N&=\underbrace{A}_{e^c}e^{-\lambda t} \end{aligned} $$
$$ \frac{\text{d}y}{\text{d}x}+P(x)y=Q(x) $$
$$ I=e^{\int P(x)\,\text{d}x} \\ y(x)=\frac{1}{I} \int Q I \,\text{d} x $$
Example:
$$ \begin{aligned} \frac{\text{d}y}{\text{d}x}-y&=x^2 \\ I=e^{\int -1\,\text{d}x}&=e^{-x} \\ y=\frac{1}{I}\int QI\,\text{d}x&\frac{1}{e^{-x}}\int x^2e^{-x}\,\text{d}x \\ y&=-x^2-2x-2+ce^x \end{aligned} $$
