# Tagged Questions

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### Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
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### Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$?

Can we write $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds\tag 1$ ? In other words, is this result valid? If so, could you help me to get the proof it NB :: ...
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### Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
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### Solving second order differential equation numerically with values given at intermediate points.

I need to numerically solve the equation, $$y''(x) + p(x)y(x) = 1$$ in the range [a,b] with conditions \begin{eqnarray} y'(\alpha) &=& 1\\ y(\beta) &=& 0 ...
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### Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
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### System of ODE - Solution

I have a system of ODE to solve $$A_{5 \times 5}\ddot{q}(t)_{5 \times 1}+ B_{5 \times 5}\dot{q}(t)_{5 \times 1}+ C_{5 \times 1} =0\tag 1$$ Given Data $A,B,C$ are constants.We know what is ...
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### Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
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### Initial value $\left ( \frac{dy}{dt} \right )+3y=11$, $y(0)=1$

I have never done an initial value problem, and would like some help on how to start this please.
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### Matrix - Commutative property

I have a rotation matrix represented as $R(t)=e^{B(t)},\tag 1$ where $B(t)$ is a skew symmetric matrix (since any rotation matrix can be expressed as a matrix exponent of a skew symmetric matrix), ...
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### Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
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### How to solve $(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$

Solve $$(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$$ I have a very similar problem like this on my homework, and I have no clue how to set it up or even start. How could I set ...
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### Why does solving $\int \frac{v}{9.8-0.0025v^2}\mathrm{d}v=\int1{d}x$ for $v^2$ in terms of $x$ produce 2 completely different answers?

In this question $g=9.8$ (acceleration of free fall). You are also given that when $x=0$ $v=0$. My answer is $v^2=400g(1-e^\frac{x}{200})$. I obtained it by integrating both sides so that ...
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### Integral of $\exp(-x\,f(x))$

What is the evaluation of the integral of the following form or is there any alternative form for it? $$\int e^{-x \, f(x)} dx \tag 1$$
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### Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
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### ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
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### Integration of nonlinear and linear ODEs

$$\frac{dc_1}{d\tau}= \alpha I(1-c_{0}) + c_{1} (-K_{F} - K_{D}-K_{N} s_{0}-K_{P}(1-q_{0}))+ c_{0}(-K_{N} s_{1}+K_{P}q_{1}), \nonumber$$ ...
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### Differential equation which has following solution $y=\frac{1}{1+\exp(ax)}$

Is there any linear differential equation which has following solution $$y=\frac{1}{1+\exp(ax)}$$ $a$ is constant. something like: $$y'' + by' +cy + \alpha = 0$$ where $b$, $\alpha$ and $c$ are ...
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### Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
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### Solving an ODE using variations of parameters and Wronskian theorem.

So I am attempting to solve this differential equation by trying to follow an example that my professor did in class. I am just not too sure about my answer seeing as WolframAlpha gives me this: ...
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### Initial Value Problem: $\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1$

Initial Value Problem: $$\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1$$ I know I'm supposed to separate the values and integrate. this is where I get stuck: $$y+\ln y = -x\cos x+\sin x+c$$ This ...
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### Implementing Equation on current data

I am trying to implement Personality, Gender, and Age in the Language of Social Media equation. I have 5 patterns and one list of 100 text = 900 words. The result of find a Match in the 900 to the ...
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### Differential equation with sec

With $(a)$ I got that $-y^2 dx = \sec^2x\ dy$, but it makes no sense. Hence, no Idea how to handle $(b)$.
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### Calculate the volume of water in glass over time.

For A) I found that volume should be defined by But I got no idea what to do in b) and c)
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### On the constant of integration in solving ordinary differential equations

I very much suspect this but I'm not sure if it's correct: In solved differential equations, does the constant 'c' always represent the value of the dependent variable when the independent=0 ?