Special Functions Lecture 1

\[\begin{aligned} \int \, x + \sqrt{x} \; dx &= \frac{x^{2}}{2} + \frac{2x^{\frac{3}{2}}}{3}+c \\ \int \, \sin^{3}(2x)\cos(2x) \, dx &= \frac{1}{2}\,\frac{sin^{4}(2x)}{4} +c\end{aligned}\]

In general: \[\begin{aligned} \int \, f^{n}(x)f^{'}(x)\,dx &= \frac{f^{n+1}(x)}{n+1} + c \qquad where\ n\neq-1\\\\\end{aligned}\] In case \(n = -1\) \[\begin{aligned} \int\,\frac{f^{'}(x)}{f(x)}\,dx &= \ln|f(x)|+c\end{aligned}\]

\[\begin{aligned} f^{-1}(f(x)) = f(f^{-1}(x)) = x\end{aligned}\]

Important Example: \[\begin{aligned} e^{ln\,f(x)} = \ln\,e^{f(x)} = f(x)\end{aligned}\]

**Ordinary differential equations** :

equations relating an independent variable (e.g. \(\displaystyle x\)) with a dependent variable (e.g. \(\displaystyle y\))

and the derivatives of of the independent variable with respect to the independent variable (e.g. \(\displaystyle \frac{dy}{dx} \) , \(\displaystyle \frac{d^{2}y}{dx^{2}} \) )

with allowing the absence of \(x\), \(y\), or both.

with forms like : \[\begin{aligned}
&f(x,\ y,\ y^{'},\ y^{''},\ y^{'''},\cdots\ y^{(n)})\\
&f(y,\ y^{'},\ y^{''},\ y^{'''},\cdots\ y^{(n)}) \qquad note\ the\ absence\ of\ x\\
&f(x,\ y^{'},\ y^{''},\ y^{'''},\cdots\ y^{(n)}) \qquad note\ the\ absence\ of\ y\\
&f(y^{'},\ y^{''},\ y^{'''},\cdots\ y^{(n)}) \qquad note\ the\ absence\ of\ x\ and\ y\end{aligned}\] **The order** of an ODE is the highest derivative present in the equation

**The degree** of an ODE is the power of the highest derivative present in the equation **Solving ODE** is getting a relation between the dependent variable (e.g. \(y\)) and the independent variable (e.g. \(x\)) without the derivatives

and it yields a general solution with as many unknown constants as the **Order** of the ODE, and that is called a **general solution**, given initial values for a particular case allows us to solve for the constants and get the **particular solution** of the ODE

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