-2
votes
0answers
13 views

Show that a vector field is C^1 . [on hold]

If you are given a system of nonlinear differential equations, say x' = f(x,y) and y'=g(x,y). How does one go about showing that f(x,y) and g(x,y) have continuous first derivative? Can we just take ...
1
vote
1answer
22 views

Show that $ y''+\tan y\cdot(y')^2=0$ when $y=\tan^{-1}(\sin h x)$

I have tried solving it in this manner: $$\tan y=\sin h x$$ Differentiate with respect to $x$: \begin{align*} \sec^2y_1&=h \cos hx\\ sec^2 y\frac{d y}{dx}&=h \cos hx\\ \frac{d ...
0
votes
0answers
42 views

Second order differential equation, orthogonality

A temperature field T(x, t) is determined by the following governing equation: $$\frac 1\alpha\frac {dT}{dt} = \frac {d^2T}{dx^2}$$ (Eq 1) T(x,t) can be expressed as a form of expansion of T(x,t) = ...
1
vote
2answers
40 views

Integrable combinations - I can't seem to arrive at the given answer

I need help! I can't seem to arrive at the answer given in our textbook. I'm new here, so I really need help. The instruction says that I need to solve this D.E by recognizing integrable ...
0
votes
1answer
20 views

Laplace transform on a non-standard sort of problem

I don't know where a laplace comes into play here: $\ddot{a}+2a=0,a(0)=b_1,\dot{a}(0)=b_2$ I am meant to solve the above using a Laplace transform, but I don't see how I would use it here? I ...
0
votes
1answer
25 views

Show all solutions of an ODE tend toward a single limit as $t$ approaches infinity and find the limiting value

Given the ODE $2y'+ty=2$ Show that all solutions approach a limit as $t\to \infty$. I've found \begin{align} y=\exp(-t^2/4)\int_0^t \exp(s^2/4) \mathrm{d}s + Ce^{-t^2/4} \end{align} However, I am ...
1
vote
2answers
59 views

Looking for a matrix A(t)

I need your help, I'm looking for a contraexample, I need to give a matrix A(t), such that $$e^{\int_0^tA(s)ds}$$ is not a matrix solution for $x'=A(t)x$. I really don't have any clue what can it be. ...
2
votes
2answers
48 views

Prove second derivative of $g$ is proportional to $g^2$

From Apostol's Calculus Vol. 1, chapter 6.26, exercise 30: Let $f(x) = \int_0^x (1+t^3)^{-1/2} dt$. $a)$ Prove $f$ is strictly monotonic. $b)$ Let $g$ be the inverse of $f$. Show that the ...
0
votes
2answers
32 views

Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
1
vote
2answers
31 views

Inverse Trigonometric functions - Boyce & Diprima 2.2.19

The problem asks for a solution to the initial value problem: \begin{align} &\sin(2x)dx+\cos(3y)dy=0\\ &y\left(\frac{\pi}{2}\right)=\frac{\pi}{3} \end{align} The problem is separable and I ...
1
vote
2answers
53 views

Simple Derivative paradox

Suppose I define $y(x)=x^3$ $${dy(x) \over dx} = 3x^2$$ $${dy(x) \over dy} = 1 = 3x^2 \frac{dx}{dy} = 0\text{ since }x \neq f(y)$$ $1 \neq 0$ If you take the differential $d()$ where $dy(x)$ then ...
2
votes
1answer
46 views

solve this ordinary differential equation?

i have the differential equation $y'=\frac{y-x}{y-x+1}$, how i solve this? try: i tryed to substitute $u=y-x$, then $u=y-x\iff y=u+x\Rightarrow y'=u'+1$ then $y'=\frac{y-x}{y-x+1}$ become ...
1
vote
0answers
31 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
0
votes
1answer
24 views

Growth equations

Year 2003 there was approx. 10 % of a substance, the year 2013 the substance is 40 % One modell which can describe the substance speed of growth is that the substance, increases every moment, is ...
0
votes
0answers
31 views
+50

Calculating the constants in the general solution of second order homogeneous ODE reduced from Riccati equation

I am writing a code to simulate a kind of volumetric flow, and I have encountered the non-linear Riccati equation in its general form near the end of my calculations. I am having trouble finding the ...
0
votes
2answers
45 views

How to find the derivative of the flow of an autonomous differential equation with respect to $x$

Ok, may be this is a silly question but consider the following. Let $\dot x=f(x)$ be an autonomous differential equation with $f$ having enough smoothness (Say $C^2$). Let $\xi:\mathbb ...
0
votes
0answers
16 views

Constant solutions of separable ODE

Consider the IWP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 $$ for continuous functions $g : I \to \mathbb R$ and $h : U \to \mathbb R$ on open intervals $I, U$ with $(x_0, y_0) \in I\times ...
2
votes
1answer
90 views

Differential equation $x\dfrac{dy}{dx}-xy=y^2$

How do I rearrange this ODE so that I can use the basic ODE techniques to solve it i.e. separable, first order using integrating factor and/or Exact ODE. $x\dfrac{dy}{dx}-xy=y^2$
1
vote
3answers
46 views

Finding solutions of $y'''-4y''+5y'-2y=-x^2+5x+2$

Find all solutions of $$y'''-4y''+5y'-2y=-x^2+5x+2.$$ I know how to find the solutions of the corresponding homogenous differential equation $y'''-4y''+5y'-2y=0$. I've done that in the following way: ...
2
votes
1answer
49 views

Deriving the equations of motion. Finding the critical points and determining their nature.

I have Duffing equation as $\ddot{x} - x + x^3 = 0$ Which I know describes the motion of a mechanical system in a twin well potential. I have let $y_1 = x, y_2 = \dot{x}$ and I want to derive the ...
0
votes
1answer
45 views

Linearizing a nonlinear system of ODE about an equilibrium

Since the method below is probably correct, and correctness is potentially irrelevant to my ability to do what I want to learn. Assume below is correct. ...
0
votes
1answer
39 views

Linearizing systems about critical points.

$$\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}\def\l{\lambda}\def\f{\frac{\sqrt{11}}{2}}$$ Find all the critical points of the following systems and derive the linearised system about each ...
5
votes
2answers
103 views

solution of $y' + y^2 = \varphi^2(x)$

I need to solve differential equation in the interval $[-\pi/2,\pi/2]$ \begin{eqnarray} y''(x) = y(x)\sin^2x \end{eqnarray} Trying $y(x) = \exp(\psi(x))$ yields, \begin{eqnarray} \zeta'(x) + ...
0
votes
3answers
41 views

Show that $u(t) \leq u(a) e^{\int_a^t f(s) ds}$

Let u(t) be a continuously differentiable function on [a,b] and the following inequality holds $\forall t \in [a,b]$ such that $u'(t) \leq f(t)u(t).$ Show that $u(t) \leq u(a) e^{\int_a^t f(s) ds}$ I ...
0
votes
1answer
29 views

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 :: ...
1
vote
1answer
76 views

Transforming ODEs into exact equations.

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 ...
0
votes
1answer
23 views

Baby version of Sturm Comparison Theorem

In Problem 15-32 of Spivak's Calculus, 4th edition, he proves the following: Suppose $\phi_1$ and $\phi_2$ satisfy $$\phi_1''+g_1\phi_1=0, \\ \phi_2''+g_2\phi_2 = 0,\\[10pt] g_2>g_1, \\[10pt] ...
1
vote
1answer
43 views

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 $\frac{dy}{dx} = .75 -.005y$ ...
1
vote
3answers
53 views

Solve $y' = x^4y+x^4y^4$

Solve the differential equation $$y' = x^4y+x^4y^4.$$ I'm not sure how to deal with the $x^4y^4$ term. So far I have only encountered differential equations where the exponent of $y$ was at most one. ...
5
votes
3answers
85 views

Initial value problem for 2nd order ODE $y''+ 4y = 8x$

How can I go about solving this equation $y''+ 4y = 8x$? Progress I found the general solution for its homogeneous form. What I don't know is how to find its particular solution.
2
votes
1answer
62 views

What is wrong with this separation of variables?

I know a number of ways of solving this basic DE: $\ddot{u} = -u$ Besides the fact that the solution is obvious, one can do: $\ddot{u} = \frac{d\dot{u}}{dt} = \frac{d\dot{u}}{du}\frac{du}{dt} = ...
0
votes
5answers
88 views

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 ...
2
votes
1answer
28 views

Finding a solution basis

Find a real solution basis of $$y'=\left( \begin{matrix}-1&-2&0\\0&2&0\\-1&-3&2\\ \end{matrix} \right)y.$$ The characteristic equation of this matrix is $$P(t) = ...
2
votes
1answer
53 views

Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
1
vote
1answer
78 views

Show f is not differentiable at x=0

(c) {22 markes} Let $$ f({\bf x}) = \begin{cases} \dfrac{x_1 x_2^2}{x_1^2 + x_2^2} & : {\bf x} \ne {\bf 0} \\[1ex] 0 & : {\bf x} = {\bf 0} \end{cases} ...
0
votes
2answers
78 views

Is $\dfrac {dy} {dx} = \dfrac {2x} {3y}$ a homogeneous differential equation?

I have a differential equation $\dfrac {dy} {dx} = \dfrac {2x} {3y}$ whose solutions are $y = \pm \sqrt {\dfrac 2 3}x $ which when I back-substitute I get $LHS=RHS$. From the definition on ...
1
vote
1answer
71 views

Non linear ordinary differential equation

How to solve the ordinary differential equation $\frac{d^2y}{dx^2}+\sin(x+y)=\sin x,y(0)=0,y'(0)=1$ Then its possible to solve it by numerical methods?
0
votes
2answers
19 views

Polynomial division to function multiplication on ODE with Separable Method

I want to solve the ODE below with the Separable Method. I know I need to see the product of $f(y)$ and $f(x)$, but I don't remember the algebra needed to see it on the polynomial division: ...
1
vote
6answers
76 views

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.
1
vote
1answer
48 views

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), ...
4
votes
5answers
76 views

Integrate $\int \left(A x^2+B x+c\right) \, dx$

I am asked to find the solution to the initial value problem: $$y'=\text{Ax}^2+\text{Bx}+c,$$ where $y(1)=1$, I get: $$\frac{A x^3}{3}+\frac{B x^2}{2}+c x+d$$ But the answer to this is: ...
0
votes
0answers
30 views

Square a linear ODE

Assuming that I have a linear ODE without any singularities over the complex numbers $$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$ Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...
1
vote
3answers
155 views

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 $r$.
2
votes
4answers
50 views

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 ...
0
votes
1answer
29 views

Matrix Solution

I have matrix integral equation of the following form ${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ . All dimensions are indicated in equation itself. " ' " ...
1
vote
0answers
62 views

Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
0
votes
0answers
46 views

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? $$\int e^{-x \, f(x)} dx \tag 1$$
2
votes
4answers
83 views

How to solve this IVP?

Could you please help me solve this IVP? A certain population grows according to the differential equation: $$\frac{\mathrm{d}P}{\mathrm{d}t} = \frac{P}{20}\left(1 − \frac{P}{4000}\right) $$ and the ...
2
votes
0answers
38 views

Finding a solution basis of differential equation

Find a solution basis of $$y'=\left[ \begin{matrix}3&-4&-2\\2&-3&-2\\0&0&1\\ \end{matrix} \right]y \,\text{ and find the solution } \Phi \text{ with } \Phi(0) = (1,1,1).$$ I'm ...
0
votes
0answers
37 views

A New Take On The Snow Plow Problem

The problem: One day it started snowing at a heavy and steady rate. A snowplow started out at noon, going 2 miles the first hour and 1 mile the second hour. What time did it start snowing? I know ...