Advanced Math

Question : Apply LU decomposition method to solve 2x+4y+2z=1,x+5y+2z=1,4x-y+9z=2 Answer : Writing system in the Matrix AX= b\[\left[\begin{matrix}2&4&2\\1&5&2\\4&-1&9\\\end{matrix}\right]\left(\begin{matrix}x\\y\\z\\\end{matrix}\right)=\left(\begin{matrix}1\\1\\2\\\end{matrix}\right)\]To find LU decomposition of A\[A=\left[\begin{matrix}2&4&2\\1&5&2\\4&-1&9\\\end{matrix}\right]\]Step 1A=LUWhere L is lower triangular matrix and…

Question : Prove that every cyclic group is abelian. Answer : Cyclic Group : A group G is said to be cyclic if G has a generator. Abelian Group :…

Question : What is example of noncommutative group ? Answer : Noncommutative Group : A group (G , *) is called noncommutative if there exists some A and B in…

Question : what is example of a nonabelian group? Answer : Nonabelian Group : A group (G , *) is called nonabelian if there exists some A and B in…

Question : How many ways are there to sit 5 boys and 11 girls at a circular table so that no two boys sit next to each other? Answer :…

Question. solve Sturm Liouville Problem y’’-λy=0, y(0) = 0, y’ (1) = 0 where λ ∈ R Solution: To solve \[y’’ - λy = 0\] (1) \[ y(0)\ =\ 0,\…

Root test examples Example (i) By the use of Root test determine whether the following series are convergent or divergent \[ \sum_{n=1}^{\infty} \frac{3^n}{n^n}\] Solution By the use of Root test…

slope field examples (i) use the slope field to guess the form of the solution curves of the differential equation dy/dx=-x/y (ii) Using the slope field, guess the form of…

Examples of Fourier series Example 1 :Solution : Fourier series of \[f\left(t\right)=t+\pi,\ -2\pi<t<2\pi\] is given as \[f\left(t\right)=\frac{a_0}{2}+\sum_{n=1}^{\infty}{a_n\cos(nt)}+b_n\sin(nt)\] Where \[a_0=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}f\left(t\right)dt=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}{(t+\pi)dt}=\frac{1}{2\pi}4\pi^2=2\pi\] \[a_n=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}f\left(t\right)dt=\frac{1}{2\pi}\int_{-2\pi}^{2\pi}{\left(t+\pi\right)\cos{\left(nt\right)}dt}=\frac{1}{2\pi}2πsin2nπ]n=sin2nπ]n\] Since \[\sin{\left(2n\pi\right)}=0\ for all n = 1,2, 3,4,5,…\] \[a_n=0, n…