## Transforming ODEs into exact equations. November 26    1

I want some examples of ODEs that can only be solved by transforming them into exact equations. They shouldn't be solvable with; Direct integration, separation of variables, manipulating a reverse product rule, homogenous linear equation method, Bern

## How to classify equilibrium points November 26    1

I have the two differential equations:$$\frac{dN_1}{dt} = N_1(2 - N_1 - 2N_2)$$ $$\frac{dN_2}{dt} = N_2(3 - N_2 - 3N_1).$$I worked out the equilibrium points to be at $N_1 = 0, \frac{4}{5}$ ...

## Compound Growth November 26    1

This is along the same lines as the population growth problem but a little more involved. I'm really lost on how to go about it. The question is: Suppose that each time a savings bank compounds the interest, you make an additional deposit $D(t)$. Sho

## Continually Compounded Interest + Addition to Principal November 26    1

This is essentially the continually compounded version of this question.I want to know how much money I will have after continually compounding interest, plus continually adding a fixed amount to the principal.Let t be time in years, S be amount save

## Is the linear combination of two solutions of a nonhomogeneous differential equation also a solution November 26    1

The question reads, if y1 and y2 are solutions of:$y''+x^2y'-e^xy=1$then is any linear combination of y1, y2 also a solution.I know for a fact that the above statement is true for homogeneous equations; however does it still hold for the nonhomogeneo

## solutions to nonhomogeneous system of differential equations with general solution already known November 26    2

Let's say we have the general solution to $X' = A(t)X$, where $X=(x_1, x_2)^T$. How do you find the general solution to the system $X'= A(t)X + b(t)$ where $b(t)$ is a $2 \times 1$ matrix with two polynomials as entries. How do you find the particula

## Getting wrong answer with reduction of order November 26    1

I always mess up reduction of order.$y''-4y'+4y=0$ has solution $y_1=e^{2x}$ using reduction of order $$y_2=y_1 \int \frac{e^{-\int{p(x)dx}}}{y_1^2}dx=e^{2x} \int \frac{e^{-(-4x)}}{(e^{2x})^2} dx=e^{2x} \cdot e^{2x}+C=e^{4x}+C$$but the answer key giv

## reduction of order on nonhomogenious ODE November 26    1

Use reduction of order to find the homogeneous and associated particular solution. $y''-3y'+2y=5e^{3x}$ given $y_1=e^x$For homogeneous: $y_2=y_1 \int \frac{e^{-\int P(x) dx}}{y_1^2}dx=e^x \int \frac{e^3x}{e^2x} dx=e^{2x}$ Now what do I do?for a parti

## Reduction of order and lost in arithmetic November 26    1

$4x^2y''+4xy'+(4x^)y=0, x>0, y_1(x)=\frac{1}{\sqrt(x)}sinx$Okay, I am so lost in the arithmetic of this problem. $y_2(x)= u(x)\frac{sinx}{\sqrt(x)}$ obviously, but I get so lost trying to find the next differential equation. I figure there's no po

## Using Reduction of Order With Second Linear DEs November 26

If I have the characteristic equation $$ar(r-1)+br+c$$ with a repeated root r1, how can I use the method of reduction of order to show that $$y=x^{r1}(c_{1} + c_{2}ln(x))$$ is the general solution of the DE?I know that one solution is $$x^{r1}$$I

## General solution to differential equations using Frobenius method November 26    1

$$x^2y''-2xy'+(x^2+2)y=0$$ The solution for the first indicial root is, $$y_1=a_0\cos x+a_1\sin x$$ and the solution for the second indicial root is, $$y_2=a_0\frac 1x \sin x+2a_1(\frac 1x \cos x-\frac 1x)$$ can a generalized solution be formed such

## Comparing numerical methods given a system of nonlinear first-order ODEs November 26

I have a system of nonlinear first-order autonomous IVP ordinary differential equations for which I'll solve numerically since I can't obtain a closed-form solution. What are the notions that matters most when comparing candidate numerical methods in

## Determining an algebraic equation for $\gamma'$ from differential equation. November 26    1

I'm attempting to determine an algebraic equation for $\gamma'$ over $[0,1]$ from Equation (1) below; however, I'm having difficulty doing so. This is in relation to a shear-thinning generalised Newtonian fluid problem (or Carreau's Law, not that thi

## Streamlines tangent to velocity field November 26    1

As from the title, I'm not too sure how they are related. Definition is that streamlines are instantaneously tangential to the velocity vector of the field. Why would a steamline that shows direction be a tangent to the velocity? Thanks!The velocity

## Zero velocity field inside an ellipse November 26    1

I'm investigating the velocity field induced by a continuous distribution of 2D vortex points distributed along an ellipse $\{a\cos\theta,b\sin\theta\}$. I'm interested in the field inside the ellipse, and I need some help to prove whether this field

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