Differential Equations

Dear community,I would like to analyze how the following function changes, when variables A,B, or C are not zero:$f(A,B,C) = \frac{Q(1+X)(PA-VB)-FC}{GX}-\frac{A(1+X)(QP(X+A+AX)-QV(X+B+BX)-FC)}{GX(X+A+...

1 hours limit Start 2:40Content:  •7.1 Modeling Situations with Differential Equations • 7.2 Verifying Solutions for Differential Equations • 7.3 Sketching Slope Fields • 7.4 Reasoning Using Slope Fie...

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1st year undergraduate  textbook for reference: Mathematics for economists: An introductory textbook (new edition): An Introductory Textbook

Please only answer questions 1,2,4 and 6. 

What is the volume of water in each bucket in a system of leaky buckets with the following conditions: There are $N$ buckets with $W_i(t)$ volume of water. A bucket has infinite capacity, but a finit...

Differentiate $y=((e^x)-(e^{-x}))/((e^x)+(e^{-x}))$  and   prove that $dy/dx=1-y^2$.

a) Given the differential equation: $\frac{du}{dx} + u = 3$ , what is the general solution?b) Given another differential equation $\frac{d^{2}y }{dx^{2} } + \frac{dy}{dx} - 6y = 0$ , what is the gener...

I have been struggling with this for over 2 hours and nothing seems to work, some points that hinder me are - if y* is a critical point, and thus the derivative is 0, then the derivative of the deriva...

Derive the weak formulation of the partial differential equationwith the initial value $u(x,0)=u_0(x)$.Please provide a detailed ans easy to follow solution. 

Answer all 5 questions with all the work shown.

Prove that the system of ODEs $y_1'=-y_1^3 + y_2^5$ $y_2'=-y_1y_2^4-y_2^3$ is asymptotically stable in $(0,0)$.

We have the system of ODEs $y_1'= y_2$ $y_2'=-y_2-\sin(y_1)$. Decide for each of the following functions whether it is a Lyapunov-function of $(0,0)$ or not: $V(y_1,y_2)= y_1^2+y_2^2$. $V(y_1,y_2)= \f...

We have the system of ODEs $y_1'=y_2-y_1 \cdot f((y_1,y_2)^T)$ $y_2'=-y_1-y_2 \cdot f((y_1,y_2)^T)$, where $f \in C^1(\mathbb{R}^2, \mathbb{R})$. Prove the following statement: If $f((y_1,y_2)^T)>0...

Find all real solutions of the linear equation $y''+\frac{4}{x}y'-\frac{10}{x^2}y =0$ on the interval $(0, \infty)$ by using the substitution $y(x)=z(\ln(x))$.    

Find the $\textbf{general}$ solution of $y'= \left (\begin{matrix} 1 & 1 & 1 \\ -1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right ) \cdot y$  and a solution with the initial value  ...

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Let $\alpha \in \mathbb{R}, \alpha > 0$. Prove that the IVP $y'=|y|^{\alpha}, \quad y(0)=0$ has a uniqe solution on $[0, \infty)$ if and only if $\alpha \geq 1$.

Find an explicit formula for a function $u$ solving the initial value problem \begin{cases} u_t+b \cdot Du+cu=0 \hspace{0.5cm} & \text{in } \mathbb{R}^n \times \left(0,\infty \right) \\ u=g \hspa...

Suppose $u \in C^2(\R^n)$ is a harmonic function. Prove that $v=|\nabla u|^2$ is subharmonic, i.e. $-\Delta v \leq 0$

Partial Differential Equation Method to be used: Symmetries Equation: $U_t*U_{x}^{2}-U_{xx}=0=F $        is         $U_t*U_{x}^{2}=U_{xx} $Has an infinite dimensional algebra of infinitesimal point sy...

Let N(t) represent the number of humans on Earth at year t.We shall assume that the population is continuously changing.Suppose the number of humans on Earth on the first day of the year in 1960 (whic...

Find the solution of the Burgers’ equation    

Need help with the following 2 questions