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Advanced Calculus

Chapter 2 · Series & ODEs

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Separation of variables
dydx=f(x)g(y)⇒∫dyg(y)=∫f(x) dx\frac{dy}{dx}=f(x)g(y)\Rightarrow\int\frac{dy}{g(y)}=\int f(x)\,dx
Exponential growth/decay
dydx=ky⇒y=Aekx\frac{dy}{dx}=ky \Rightarrow y = Ae^{kx}
Constant AA fixed by the initial condition.
Maclaurin series
f(x)=∑n=0∞f(n)(0)n!xnf(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n
Standard series
ex=∑xnn!,  sin⁡x=x−x33!+⋯ ,  cos⁡x=1−x22!+⋯e^x=\sum\frac{x^n}{n!},\;\sin x=x-\tfrac{x^3}{3!}+\cdots,\;\cos x=1-\tfrac{x^2}{2!}+\cdots
ln(1+x)
ln⁡(1+x)=x−x22+x33−⋯\ln(1+x)=x-\tfrac{x^2}{2}+\tfrac{x^3}{3}-\cdots
Valid for −1<x≤1-1<x\le 1.
Maclaurin
f(x)=∑f(n)(0)n!xnf(x)=\sum \tfrac{f^{(n)}(0)}{n!}x^n
Key expansions
ex=∑xnn!,  sin⁡x=x−x33!+⋯e^x=\sum\tfrac{x^n}{n!},\;\sin x = x-\tfrac{x^3}{3!}+\cdots
Separable ODE
∫dyg(y)=∫f(x) dx\int\tfrac{dy}{g(y)} = \int f(x)\,dx
Separate variables, integrate both sides, +C from initial condition.
Common trap
Apply the initial condition to find C; Euler’s method is an approximation (step size matters).