Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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Particular solution to a Riccati equation $y' = 1 + 2y + xy^2$

The equation is $y' = 1 + 2y + xy^2$. I've tried $mx+n$, $ax^m$, even $\tan x$ as candidates for particular solution where $a,m,n \in \mathbb Q$, but it did not work. Can anyone find one particular ...
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3answers
2k views

Integrating with respect to different variables

I have started reading a book on differential equations and it says something like: $$\frac{dx}{x} = k \, dt$$ Integrating both sides gives $$\log x = kt + c$$ How is it that I can ...
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3answers
540 views

How to solve the following differential equation: $(2x^3y^2-y)dx+(2x^2y^3-x)dy=0$?

I'd love your help with solving this following differential equation: $$(2x^3y^2-y)dx+(2x^2y^3-x)dy=0.$$ I tried to use check if this is an exact equation and find a integration, but it didn't work. ...
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4answers
310 views

What is the quickest way to solve this 2nd Order Linear ODE?

This appeared on my professor's test review, and its taken me hours to, surprise surprise, get the wrong answer. Could someone help me with the method I should be using to solve this? ...
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2answers
2k views

Determine the *interval* in which the solution is defined?

The ODE: $y' = (1-2x)y^2$ Initial Value: $y(0) = -1/6$ I've solved the particular solution, which is $1/(x^2-x-6)$. I don't understand what they mean about the solution is defined, because when ...
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5answers
135 views

Differential Equation Question $y'=\frac{1}{\cos y-x}$

I want to find the solution for the first order differential equation $$y'=\frac{1}{\cos y-x}$$ I have no clue what to do, what I tried is: $$(\cos y-x)dy=dx$$ $$\frac{dy}{dx}=\frac{1}{\cos y-x}$$ ...
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2answers
1k views

Can a differential equation have non unique solutions?

There are theorems of existence and uniqueness of differential equations. I was wondering if it is possible that a differential equations has a solution but it is not unique.
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3answers
1k views

Differential Equations without Analytical Solutions

In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is: Can you prove a differential equation ...
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4answers
563 views

How can I solve this linear differential equation $y^{\prime\prime}-4y^{\prime}+3y=\frac{1}{1+e^{-x}}$?

My problem is to solve this linear differential equation: $$y^{\prime\prime}-4y^{\prime}+3y=\frac{1}{1+e^{-x}}$$ My approach was: i can see this must be an ordinary differential equation of ...
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4answers
132 views

$u''(t)+u(t) = |t|$

Solve the Cauchy problem, $\forall t \in \mathbb{R}$, $$ \begin{cases} u''(t) + u(t) = |t|\\ u(0)=1, \quad u'(0) = -1 \end{cases} $$ The solution to the homogeneous equation is $A\cos(t) + B ...
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2answers
81 views

How can i solve this System of first-order differential Equations?

My Problem is this given System of differential Equations: $$\dot{x}=8x+18y$$ $$\dot{y}=-3x-7y$$ I am looking for a gerenal solution. My Approach was: i can see this is a System of linear and ...
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2answers
103 views

Use Euler's method to approximate $\int^2_0 e^{-u^2}du$

We learned Euler's method today there is one hw problem totally stunned my hat off. It says: Use Euler's method to approximate $\int^2_0 e^{-u^2}du$. I know Euler's method is $y_{n+1} = y_n + ...
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3answers
749 views

Finding fundamental set of solution of higher order differential equation

The fundamental set of solution $$y^{(4)} - 16y = 0$$ I worked this problem out but I was under the impression that I can apply the general method of characteristic equation to solve : $$ r^4 - 16 = ...
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3answers
205 views

What are other solutions to this differential equation, “similar” to $\sin x$ and $e^x$?

I've been studying electronics, where they make great use of the relationship between the sine and exponential functions ($e^{i \omega t} = \cos{\omega t} + i \sin \omega t)$. This relationship is ...
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2answers
327 views

An interesting pattern in solutions to differential equations

OK, watch this: Suppose I have a weight on the end of a spring. Assuming the spring obeys Hooke's law, as the weight is displaced from its rest position, the spring exerts a restoring force in the ...
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3answers
532 views

Square root of differential operator

If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$. How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number) For ...
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5answers
1k views

particular solution of $y''+y'=xe^{-x}$

I'm using the method of undetermined coefficients to find a particular solution of: $$y''+y'=xe^{-x}$$ Ostensibly, it seems that $y_p$ should take the form of $(Ax + B)e^{-x}$ At least ...
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2answers
576 views

General solution for $y^{iv}+ 2y''+y=\cos x$

Here is another problem from Mathews and Walker that has given me some trouble. 1-18. Find the general solution of $y^{iv}+ 2y''+y=\cos x$. Note: Thanks, everyone, for clearing up the ...
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2answers
547 views

Why is this constant of integration taken as $\log A$ instead of just $C$?

Suppose we solve $$\frac{dy}{dx} = \frac{1 + y}{2 + x} .$$ Which can be written as the following and integrating both sides w.r.t. $y$ and $x$: $$\int\frac{1}{1 + y}dy = \int\frac{1}{2 +x}dx ,$$ we ...
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2answers
478 views

integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?

In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain: ...
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2answers
557 views

Differential equation, quite weird task

I'm having some trouble while trying to understand one task.. The task is as follows: $$\ddot{x}(t) + \dot{x}(t) + 2x(t) = \sin(\omega t)$$ where $x(0) = 7, t\geq 0$ The solution is in the following ...
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1answer
110 views

solving a differential equation involving $\frac{y-x^2}{\sin y-x}$

I'm trying to find the general solution to $$\frac{\text{d}y}{\text{d}x} = \frac{y-x^2}{\sin y-x}$$ Any ideas would be greatly appreciated. Thanks!
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4answers
145 views

Solving a challenging differential equation

How would one go about finding a closed form analytic solution to the following differential equation? $$\frac{d^2y}{dx^2} +(x^4 +x^2+x+c)y(x) =0 $$ where $c\in\mathbb{R}$
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2answers
121 views

Solving a 5 dimensional function in a neighbourhood

Consider a function $f:\mathbb{R}^5 \to \mathbb{R}^2$ defined by $$f(u,v,w,x,y)=(uy+vx+w+x^2,uvw+x+y+1)$$ such that $f(2,1,0,-1,0)=(0,0)$ (i) Show that we can solve $f(u,v,w,x,y) = (0,0)$ for ...
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2answers
145 views

Estimating rate of blow up of an ODE

Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$. Several related questions: (1) Can I ...
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3answers
281 views

Solving $f'(x) +f(x)=cf(x-1)$

To show that $f(x) =Ae^{nx}$ for constant $n$ and $A$ starting with this thing: $$f'(x) +f(x)=cf(x-1)$$ Where $c$ is constant and $c\not= 0$. If it wasn't for the $f(x-1)$ bit, I would just use the ...
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2answers
194 views

General solution of $C_{n+2}(x)=xC_n(x)+nC_{n-1}(x)$

Airy differential equation. $y''(x)=xy(x)$ $y'''(x)=y(x)+x y'(x)$ $y'^v(x)=x^2y(x)+2 y'(x)$ $y^v(x)=4xy(x)+x^2 y'(x)$ $y^{(6)}(x)=(x^3+4)y(x)+6x y'(x)$ . . $y^{(n)}(x)=A_n(x)y(x)+B_n(x) y'(x)$ ...
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1answer
335 views

Sloppy notation for differential equations

Why does one often use the following notation for differential equation: $$ y'=f(t)y$$ (this is just a particular example) ? What bothers me with this notation, which I have encountered in ...
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2answers
167 views

how to solve $x^3y′′−xy′+y=0$

I tried to use Frobenius method to solve $$ x^{3}{\rm y}′′\left(x\right) − x\,{\rm y}′\left(x\right) + {\rm y}\left(x\right)=0, $$ but it does not work. And the solution most be $y_{1} = ax + b$. I ...
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2answers
108 views

Solve differential equation $y'' = -a y +\frac by$

I am trying to solve for y(t): $$y'' =-ay + \frac by$$ I have tried a lot, but haven't succeeded so far. Actually I am not sure there is a 'nice' solution. Do any of you have ideas of how to solve ...
5
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2answers
107 views

“Constrained” numerical solutions of ODEs with conservation laws?

Hi know little about numerical methods and I was considering the following problem that possibly has standard solution in the literature. Suppose you have an ODE for wich we already know that it must ...
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3answers
417 views

Physical meaning behind Frequency domain?

I understand its usage and why is it important because It transforms differential equations to algebraic ones.. But I can't get the physical meaning of the new form of the equation and the meaning of ...
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2answers
135 views

General solution to $\frac{df}{dx}=3f$

I am learning how to do calculus and was presented with an example I am struggling a bit to understand. Why does $\frac{df}{dx}=3f$ have the general solution of $f(x)=Ce^{3x}$?
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4answers
253 views

Where can I find good, free resources on differential equations?

I'd like to know if there are any good online books, lecture notes, videos, tutorials, or similar that are free to the public (on differential equations). Suggestions are welcome!
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1answer
2k views

What's the difference between explicit and implicit Runge-Kutta methods?

I have been working on numerical analysis, just as a hobby. I am only aware of the basic fourth order Runge-Kutta method in order to solve problems. When I was digging deep into it, I found there are ...
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3answers
280 views

Numerically solving ODEs-estimating the solution between the nodes?

So I have heard about a lot of fancy numerical methods for solving ODEs. I know that there are methods that give asymptotically a low error like the Runge-Kutta methods. (Assuming sufficient ...
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2answers
269 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem: Let $f(x):\Omega\longrightarrow \mathbb R^n$ a $C^2$ function where $\Omega$ is an open subset of ...
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1answer
253 views

How to solve $y(y'+3)=ax^2+bx+c, \quad a,b,c \in \mathbb{R}$

How could we solve this differential equation $$y(y'+3)=ax^2+bx+c, \quad a,b,c \in \mathbb{R}$$ I really don't know how start. I am not familiar with this sort of differential equations (I know it is ...
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1answer
85 views

How to solve the following differential equation $\frac{dy}{dx} =\frac{x+2y+8}{2x+y+7}$

How to solve the following differential equation $\displaystyle\frac{dy}{dx} =\frac{x+2y+8}{2x+y+7}$.
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6answers
1k views

Free differential equations textbook?

I've seen questions on what are some good differential equations textbook and people generally points to Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard and so on I was ...
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296 views

The Green’s function of the boundary value problem

What is the Green’s function of the boundary value problem $$ \frac{\mathrm d^2 y}{\mathrm d x^2}-\frac{1}{x}\frac{\mathrm dy}{\mathrm dx}=1,\quad y(0)=y(1)=0, $$ this boundary problem is not self ...
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2answers
1k views

nonlinear first order differential equation

How can I find an exact solution for this problem ? Is there any technique for cubic nonlinearity as in the case of Bernoulli differential equation? $y'=x^{3}y^{3}-1\\$
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1answer
112 views

Rank of a jet bundle of a vector bundle.

I am trying to understand the jet bundles but currently I am stuck on the following questions: Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...
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3answers
137 views

Semigroup implies exponential

Let $S :[0,1]\to \mathbb R^{n\times n}$ be a continuous function which satisfies $$ S(0)=I \quad\text{and}\quad S(s+t)=S(s)S(t), $$ for all $s,t\in[0,1]$ with $s+t\in[0,1]$. (Here $I$ is the identity ...
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3answers
645 views

Converting Second Order Linear Equations to First Order Linear Equations

$\color{green}{question}$: How can following $\color{blue}{second-order~linear~equation}$ be converted into $\color{blue}{first-order~linear~equation}$? This is our second-order linear Equation: ...
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1answer
410 views

When solving an ODE using power series method, Why do we need to expand the solution around the singular point?

When solving a differential equation using series expansion method, if it has the following form : $$y''+\frac{p(x)}{x}y'+\frac{q(x)}{x^2}y=0$$ ; where $p$ and $q$ are analytic at $x_0$; if we want ...
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2answers
218 views

What's the difference between $\frac{dy}{dx}$ and $dy$?

Ok, so I was doing a substitution problem and I realized that $dy = u\ dx + x\ du$ and not $\frac{dy}{dx} = u\ dx + x\ du$ and I was wondering what the difference was between those two. My first guess ...
5
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1answer
117 views

Examples of how to calculate $e^A$

I'm trying to learn the process to discover $e^A$ For example, if $A$ is diagonalizable is easy: $$A =\begin{pmatrix} 5 & -6 \\ 3 & -4 \\ \end{pmatrix}$$ Then we ...
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1answer
64 views

How can i solve a differential equation like this one?

My Problem is: this given differential equation $$x^3+y^3+x^2y-xy^2y^{\prime}=0$$ $$(x\neq 0,\ y\neq 0)$$ My Approach was: i had the idea to bring it in this form: $$x^3+y^3+x^2y-xy^2y^{\prime}=0$$ ...
5
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3answers
541 views

Proving Newton's Binomial Theorem

So, I've done most of the problem to this point, but just cannot figure out the last piece. I may just be missing the math skills needed to complete the proof (differential equations). Problem (from ...