what is the general solution to the ODE [closed]
What is the general solution to the ODEI would like to find the general solution to the ODE:-2y'+ y = 0
View ArticleSurface integrals in spherical coordinates
If I am given a surface in spherical coordinates $(r,\theta,\varphi)$, such that it is parametrised as:$$\begin{align}r&=r(\theta,\varphi)\\\theta&=\theta\\\varphi&=\varphi\end{align}$$What...
View ArticleUsing Matrix Calculus in Backpropagation derivation. Rules of order of matmul...
I define a neural network with $L$ layers ($L-1$ hidden layers). The forward pass is as follows:$$\mathbf{a}^{(l)} = f(\mathbf{W}^{(l)}\mathbf{a}^{(l-1)}+\mathbf{b}^{(l)})$$Where $l \in [0,L]$ and...
View ArticleIntegration of $ \int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx $
Question: What is the closed form of this following integral? $\int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx.$We have $ a_{k}:=\int_{0}^{\pi/2}{x \cos^k x dx},$ then posit from the Beppo-Levi theorem,...
View ArticleExpress the integrand as a sum of partial function and evaluate the integral...
The integral is - integral x+3/2x^3-8x dxShow all the work for better understanding
View ArticleCalculate two limits related to a recursively defined sequence
I am currently studying mathematical analysis, specifically the theory of sequences, and I have come across a challenging problem in my homework that I am unable to solve. I would greatly appreciate...
View ArticleShow that $x^n$ is a continuous function
So the idea I thought of doing was by induction.$\forall\; \epsilon > 0\;\; \exists \;\delta > 0$ such that $|x-x_0| < \delta \implies |x-x_0| < \epsilon $By the definition of the limit...
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Proof of the Chain Rule of multivariable functions in Munkres' Analysis on...
In Munkres' Analysis on Manifolds, page 57 Theorem 7.1(the Chain Rule) it states: ...For this purpose, let us introduce the function $F(\mathbf h)$ defined by setting $F(\mathbf 0)=\mathbf 0$...
View ArticleFind extreme values of function cos x cos y cos(x+y)
I am not able to find critical points after finding out the partial derivatives. Also, no specific domain has been mentioned in the question. So, I am not sure how to proceed. Please help me with this.
View ArticleCalculus-problem-differential-equations
Solve the following equation:ax"+bx+c+d((c-x)/(c+x))^2=0Find whether the functions are periodic or notPlease help, I'm lost
View ArticleExistence of $1/(e-1)$ value of a continous and differentiable function...
Let $f:[0,1]\to \mathbb R$ be a continous function on $[0,1]$ and differentiable on $(0,1)$, $f(0)=0, f(1)=1$. Prove that there exists $c\in (0,1)$ so that $f(c)+\frac{1}{e-1}=f'(c)$ where $e$ is the...
View ArticleImage of a continuous function over a convex set is convex in...
Let $f:\Omega\to\mathbb{R}$ be a continuous function over convex domain $\Omega$.Is the set $$ f\left(\Omega\right)=\left\{ y:y=f\left(x\right):x\in\Omega\right\} $$Is the set $f(\Omega)$ convex...
View ArticleA closed form for $\int_0^\infty\ln x\cdot\ln\left(1+\frac1{2\cosh x}\right)dx$
Is it possible to evaluate this integral in a closed form?$$\int_0^\infty\ln x\cdot\ln\left(1+\frac1{2\cosh x}\right)dx=\int_0^\infty\ln x\cdot\ln\left(1+\frac1{e^{-x}+e^x}\right)dx$$I tried to...
View ArticleNumerical Analysis of Differential Equation with Boundary Conditions
NoteI have decided to edit this post so it is more specific and alsobecause I was incorrect the first time. I didn't wanna make anotherpost about the same subject, but if this is not allowed then put...
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find ${dy}/{dx}$ of $x\sqrt{1+y} + y\sqrt{1+x} = 0$
QuestionMy approachI tried by applying basic product rule, but could not proceed further; don't know how to eliminate the y factors in the solution. How to prove it?
View ArticleFind the singular solution of the differential equation...
Find the singular solution of the differential equation $$4xp^2=(3x-1)^2,$$ where $p=\frac{dy}{dx}.$As we know the singular solution, of a first order differential equation, is represented by the...
View ArticleContinuity of this multivariable function
Can you please tell me if my reasoning is correct or if I missed some crucial step/point?I have to study if the following function is continuous$$f(x, y) = \begin{cases} \frac{x^2-y^2-1}{1-x} &...
View ArticleConvergence radius for Cauchy multiplication series
Let $\sum_{n=1}^{\infty} a_n x^n$ with convergence radius $R_1$, and $\sum_{n=1}^{\infty} b_n x^n$ with convergence radius $R_2$.Let $R$ be the convergence radius of $\sum_{n=1}^{\infty} c_n x^n$ with...
View ArticleLine integral around the origin involving bivariate polynomials
I encountered a mathematical problem while studying mathematical analysis.The problem is as follows:Let $n\geq 1$ and consider a polynomial $R$ of degree $2n$ in two variables that is only equal to...
View ArticleSolve $\int_{-\infty}^\infty e^{i a x-i b \tanh(\frac xb)}dx=...
$$\int_{-\infty}^\infty e^{i a x-i b \tanh(\frac xb)}dx$$This is to be solved . I don't know what the answer might look like but might be some special function . What I tried is expanding the...
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