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which ode does y=sin(log(x)) satisfy?

By SK Math Expert August 23, 2021Abstract Algebra, Advanced Math, Mathematics, Multivariable Calculus, ODEs

Question : Which ode does y=sin(log(x)) satisfy? Solution : Given function \[y=sin\left(log\left(x\right)\right)\] differentiating with respect to x \[\frac{dy}{dx}=\cos{\left(\log{\left(x\right)}\right)}\frac{1}{x}\] \[x\ \frac{dy}{dx}=\cos{\left(\log{\left(x\right)}\right)}\] Again differentiating with respect to x \[\frac{dy}{dx}+x\frac{d^2y}{dx^2}=-sin\left(log\left(x\right)\right)\frac{1}{x}=\frac{y}{x}\] \[x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+y=0\] is the…

if f(x)=(x+2)/(x+3) and y=f^-1(x) then dy/dx is equal to

By SK Math Expert August 23, 2021Abstract Algebra, Advanced Math, Linear Algebra, Mathematics, Multivariable Calculus, Real Analysis

Question : If f(x)=(x+2)/(x+3) and y=f^-1(x) then dy/dx is equal to Solution : Let the Given function \[z=\frac{x+2}{x+3}\] \[xz+3z=x+2\] \[xz-x=2-3z\] x=\frac{2-3z}{z-1} This is inverse of f in terms of the…

solved find eigen values of the matrix [5 4;1 2]

By SK Math Expert August 23, 2021Abstract Algebra, Advanced Math, Linear Algebra, Mathematics, Multivariable Calculus

Question : Find eigen values of the matrix [5 4;1 2] Solution : Given matrix \[A=\left(\begin{matrix}5&4\\1&2\\\end{matrix}\right)\] eigen values of A are given by \[\left|A-\lambda I\right|=0\] Where I is an identity…

how do you find other two roots of a cubic equation if one root of the equation is given

By SK Math Expert August 22, 2021Abstract Algebra, Advanced Math, Mathematics, Multivariable Calculus

In the following question, how do you find other two roots of a cubic equation if one root of the equation is given Question : If one root of the…

If the roots of the equation x^3-5x^2-16x+80=0 are 4, -4 then third root of the equation is

By SK Math Expert August 22, 2021Abstract Algebra, Advanced Math, Mathematics, Multivariable Calculus

Question : If the roots of the equation x^3-5x^2-16x+80=0 are 4, -4 then third root of the equation is Solution ; Two roots of the equation \[x^3-5x^2-16x+80=0\] \[α=4,β=-4\] Let third…

The distinct eigen values of the matrix [1 1 0;1 1 0;0 0 0] are

By SK Math Expert August 21, 2021Abstract Algebra, Advanced Math, Linear Algebra, Mathematics, Multivariable Calculus

Question :The distinct eigen values of the matrix [1 1 0;1 1 0;0 0 0] are (1) 0,1 (2) 1, -1 (3) 0,2 (4) 1, 2 Solution : Let the…

How to prove that F=(x^2-y^2)i-(2xy+y)j is a conservative force

By SK Math Expert August 21, 2021Abstract Algebra, Advanced Math, Mathematics, Multivariable Calculus

Question : Prove that F=(x^2-y^2)i-(2xy+y)j is a conservative force Solution : To prove that Force\[\vec{F}=\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\] is a conservative force. We shall find curl of F \[ curl\left(\vec{F}\right)=\nabla\times\vec{F}=\left(\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}\right)\times\left(\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\right)\] \[=\left(\hat{i}\times\hat{i}\right)\frac{\partial\left(x^2-y^2+x\right)}{\partial…

how to Find curl of the vector F=(x^2-y^2)i-(2xy+y)j

By SK Math Expert August 21, 2021Abstract Algebra, Advanced Math, Mathematics, Multivariable Calculus

Question : Find curl of the vector F=(x^2-y^2)i-(2xy+y)j Solution : Given vector \[\vec{F}=\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\] Then curl of F is \[ curl\left(\vec{F}\right)=\nabla\times\vec{F}=\left(\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}\right)\times\left(\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\right)\] \[=\left(\hat{i}\times\hat{i}\right)\frac{\partial\left(x^2-y^2+x\right)}{\partial x}-\left(\hat{i}\times\hat{j}\right)\frac{\partial\left(2xy+y\right)}{\partial x}+\left(\hat{j}\times\hat{i}\right)\frac{\partial\left(x^2-y^2+x\right)}{\partial y}-\left(\hat{j}\times\hat{j}\right)\frac{\partial\left(2xy+y\right)}{\partial y}\] Since \[\hat{i}\times\hat{i}=\hat{j}\times\hat{j}=0\ ,\…

The work done by the force F=(x^2-y^2)i-(2xy+y)j displacing a particle in the xy plane from (0, 0) to (1, 1) along the parabola is

By SK Math Expert August 21, 2021Mathematics, Multivariable Calculus

Question : The work done by the force F=(x^2-y^2)i-(2xy+y)j displacing a particle in the xy plane from (0, 0) to (1, 1) along the parabola is Solution : Given force \[\vec{F}=\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\]…

Solved find the sum of the series (1/n+1/(n+1)+1/(n+2)+…..+1/4n as limit n approaches infinity

By SK Math Expert August 21, 2021Advanced Math, Discrete Mathematics, Linear Algebra, Mathematics, Multivariable Calculus, Number Theory, Numerical Analysis, Real Analysis

Question : Find the sum of the series 1/(n+1)+1/(n+2)+…..+1/4n as limit n approaches infinity Solution : Let the sum of the series as \[L=\lim_{n\rightarrow\infty}{\left[\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots..+\frac{1}{4n}\right]\ }\] \[L=\lim_{n\rightarrow\infty}{\left[\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots..+\frac{1}{n+3n}\right]\ }\] \[=\lim_{n\rightarrow\infty}{\frac{1}{n}\left[\frac{n}{n}+\frac{n}{n+1}+\frac{n}{n+2}+\ldots..+\frac{n}{n+3n}\right]\ }\] \[=\lim_{n\rightarrow\infty}{\frac{1}{n}\left[\frac{1}{1+\frac{0}{n}}+\frac{1}{1+\frac{1}{n}}+\frac{1}{1+\frac{2}{n}}+\ldots..+\frac{1}{1+\frac{3n}{n}}\right]\…