0
votes
1answer
17 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
35 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
50 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. ...
3
votes
2answers
59 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
55 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
83 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
26 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
44 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
77 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
75 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
68 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
15 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
73 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
44 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
74 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
28 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
136 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
47 views

Differential equation $(x-3)(\frac{dy}{dx})+y=6e^x, x>0$

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 this up?
0
votes
1answer
27 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
61 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
81 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
37 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
35 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 ...
3
votes
3answers
132 views

General solution of Differential equation

What is solution of the differential equation: \begin{aligned}x({yy'' + y'^2}) + yy' = 0\end{aligned} What i am confused about it is the treatment of $$\left(\dfrac{dy}{dx}\right)^2$$ for solving ...
0
votes
2answers
28 views

Differential of a shifted function

If I'm given the differential equation: $$\frac{d(12-24f(t))}{dt} = 5$$ How do I rearrange this so that it looks like a normal first order linear differential equation? e.g, so it looks something ...
1
vote
2answers
38 views

Find a linear differential equation for a given function

Is there any general method for finding a linear differential equation with polynomial coefficients that is satisfied by a given elementary function (or prove that no one exists)? Example: If $f(x) ...
1
vote
0answers
25 views

Help solving particular D.E

I'm going through past exams for revision and couldn't get the same answer as the markscheme for this problem. QP ...
0
votes
1answer
67 views

Two methods of finding a function $f$ such that $Mdx+Ndy=0$ on the curves $f(x,y)=c$

this problem is from my class,i did one way and got one answer,professor did it in another way and got another answer.question is:Find $f(x,y)=constant$ where differential equation is ...
1
vote
2answers
43 views

Finding solution basis of $y^{(4)}-2y'''+5y''-8y'+4y=0$

Find a real-valued solution basis of $$y^{(4)}-2y'''+5y''-8y'+4y=0.$$ The corresponding characteristic equation is $$x^4-2x^3+5x^2-8x+4=0$$ $$\iff(x-1)^2(x^2+4)=0$$ which has the zeros $1, 2i, -2i$. ...
2
votes
0answers
31 views

Special properties of bounded functions

I have a problem understanding the reasons as to why under some circumstances a term can be omitted due to it being a part of a bounded function, and I hoped to get some clarity to this here. There is ...
0
votes
4answers
85 views

Solving the differential equation $x\frac{dy}{dx}=\frac{1}{y}+y$ [closed]

Solve the differential equation $$x\frac{dy}{dx}=\frac{1}{y}+y.$$
2
votes
3answers
41 views

Solving some inhomogeneous differential equations

I am currently reviewing some differential equations and ran into a couple of problems with the problems shown below particularly in the form of the particular solution for the equations. I haven't ...
0
votes
1answer
33 views

Partial fractions where the denominator is one function

I need to solve this differential equation for x: $$ \frac{dv}{dx} = \frac{4000}{v} - 0.9v $$ Rearranging: $$ \frac{dx}{dv} = \frac{1}{4000v^{-1} - 0.9v} $$ How would I go about splitting this ...
3
votes
3answers
113 views

Differential equation with $\sqrt{xy}$

$$7\sqrt{xy} \frac{dy}{dx}=4, \quad x,y>0$$ How do I solve this equation for $y$
0
votes
0answers
42 views

ODE with multiple simple conditions $f'(x)=f(x)(Ax+D ) $

I have an ODE to solve . The main issue is,in addition to solving it I have to keep some conditions too in the solution of f(x).. I am bit confused regarding how to deal with it. Equation is given ...
3
votes
3answers
48 views

Differential Equation $\frac{dP}{dt} = kP(1-P)$

I have a question about solving this differential equation. So, the question is to solve it given that $P(0)=\frac23$ So this is what I've done so far $$\frac{dP}{dt} = kP(1-P)$$ $$ k\,dt = ...
1
vote
2answers
48 views

Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...
2
votes
1answer
78 views

Differential Equation $\frac{dy}{dt}$ = $y - t$

Given the differential equation $\dfrac{dy}{dt}$ = $y - t$ Is this equation separable? -> No it is impossible to separate this equation because we can't get $y$ alone with $dy$ and $-t$ alone with ...
2
votes
3answers
122 views

Separable differentiable equations

Which of the following is a solution to the separable differentiable equation: $$\frac{dy}{dx}=\frac{xy}{\ln y }$$ $A.\ \displaystyle e^{|x|}$ $B.\ \displaystyle e^{\sqrt{\frac{x^2}2}}$ $C.\ ...
4
votes
2answers
137 views

Solve nonlinear differential equation

Could you help me solve or give me some advice about following differential equation $$ 2(y')^2 + 3xy'y'' + 3yy'' = 0 $$
1
vote
3answers
54 views

Why does my derivation of $\mathcal{L(\frac{f(t)}{t})}$ lead to a wrong answer?

I'm trying to prove that $$\mathcal{L(\frac{f(t)}{t})(s)} = \int_s^{\infty}\mathcal{L(f(t))}(u)du$$ Here's my attempt: $$\mathcal{L(\frac{f(t)}{t})}(s)=\int_{0}^{\infty} \frac{f(t)}{t}e^{-st}dt$$ ...
2
votes
2answers
67 views

How to prove that solution to ODE in spherical coordinate is equivalent to the ODE in cartesian coordinates if it is a thin shell

Solving a diffusion-type ODE across a spherical shell, the equation is: $$\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=0\tag{1}$$ with boundary conditions $f(r_1)=f_1$ and $f(r_2)=f_2$. The solution is: ...
1
vote
0answers
30 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
2
votes
0answers
25 views

Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
0
votes
0answers
22 views

coupled heat transfer equation

I want to try to solve a strong coupling problem, I have a variable $\zeta as$ : \begin{equation} \zeta(x,y,T)=\frac{\frac{R(x,y,T)}{\sqrt{2}}-F(T)}{F(T)-E(T)} \end{equation} Where F(T) and E(T) are ...
0
votes
1answer
45 views

Continuity of $K(x,y)$ satisfying $g(x)= \int_0^1 \! K(x,y) f(y)\ \mathrm{d}x $ and $ \frac{d^3g}{dx^3} = f$

$g(x)$ is defined as the following : $$g(x)= \int_0^1 \! K(x,y) f(y)\ \mathrm{d}x $$ where $K(x,y)$ is continuous in $ 0 \leq x \leq 1 $ , $ 0 \leq y \leq 1 $, and $f(x)$ is continuous in $ 0 \leq x ...
3
votes
0answers
67 views

Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...
0
votes
0answers
14 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
3
votes
2answers
108 views

Solving $\frac{d f(x)}{dx} + f(x-1) = x^2$

Given following differential equation: $$\frac{d f(x)}{dx} + f(x-1) = x^2$$ where $ f(x)=0 $ for $x \leq 0 $. How do I find the solution for $ x \geq 0 $ ? I understand that for $ 0 \leq x \leq 1 ...