3
votes
1answer
72 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
3
votes
0answers
67 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
0
votes
0answers
77 views

Using Picard-Lindelöf Theorem to elegantly demonstrate uniqueness of an IVP

I am trying to keep this question clean and short, therefore I won't write down the entire theorem of Picard-Lindelöf here. Problem: $$y'=1+y^2 =:F(y), \ y(0)=0 $$ Find a solution on a ...
3
votes
0answers
58 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
0
votes
2answers
30 views

Initial value problem of $y' = \sqrt{|y|}(y+1)$

i'm trying to determinate the solution of the intial value problem $$y' = \sqrt{|y|}(y+1)$$ my solution was as follow applying substitution as follow let $u^{2} = y$ and $dy = 2u\ du$ $$2 \int ...
0
votes
1answer
32 views

Analysis of first order differential equation

I'm working through a question where the differential equation is $$ y^2(y'^2 -1)(3y'^2 +1) = c, \;\;\; y(0) = 0 $$ and the answer proceeds with two cases (1) $c=0 \implies y(x)=0 \vee y(x) = \pm ...
0
votes
2answers
53 views

For 2nd order linear homogenous ODE, what is the effect of changing the value of the IC for $\mathrm{d}y/\mathrm{d}x$?

Given a linear homogenous second-order ODE $$f_2(x)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + f_1(x)\frac{\mathrm{d}y}{\mathrm{d}x} + f_0(x)y = 0$$ with initial conditions $$y(x_0)=0$$ ...
-1
votes
1answer
40 views

Solving a differential equation of degree 4. [closed]

I need to solve below differential equation analytically. $y^{(4)}-y=x\sin(x)$
0
votes
1answer
76 views

Inverse Fourier Transform of the output, Y(f)

A linear system is defined by the differential equation: $$ y''(t) + 4y'(t) + 25y(t)= x(t) $$ The transfer function of this system is: $$ H(f) = \frac{Y(f)}{X(f)}= \frac{1}{(2\pi fj)^{2}+ 4(2\pi ...
2
votes
1answer
62 views

Derive a perturbation of period $2\pi$, to order $\epsilon$

I have the following problem: In the equation $\ddot{x}+\Omega^2x+\epsilon f(x) = \Gamma \cos t$, $\Omega$ is not close to an odd integer, and $f(x)$ is an odd function of x, with expansion, $$f(a\cos ...
1
vote
1answer
144 views

How to linearize a nonlinear ODE around its equilibrium?

I am studying for a comprehensive exam in non-linear ODE's and I have this in my book: $$\ddot{\xi}+c\bigg[x_1+\xi-\dfrac{\lambda}{a-x_1-\xi}\bigg] = 0$$ then it goes straight to ...
0
votes
1answer
72 views

How to decide whether PDE is Homogeneous or non-homogeneous.

I am studying second order PDE. And I have seen homogeneous and non-homogeneous PDE. But I cannot decide which one is homogeneous or non-homogeneous. For examples; (1) $(D^3-3D^2D'+4D'^3)u=0$ ...
1
vote
1answer
53 views

Please explain. Really I dont understand and I need to learn. Pde: : example of finding particular integral

When we look at the solution part, there is a statement The PI of the given PDE is obtained as follows After the statement, I dont really understand all of the calculation. Espacially, After the ...
1
vote
1answer
277 views

How to write Pfaffian differential equation

I am studying the example. And I dont understand how to write the pfaffian diff equation in the first line. What is its formula?
3
votes
2answers
196 views

Show that the curves are circle.

For example, this question is also similar to the previous question I have asked, which is the following link; How to show the curves are conics. Question: Solve the equation ...
2
votes
1answer
55 views

Help with understanding a proof on Ordinary Differential Equations

I have this theorem in my book: Suppose that $f$ is continuous in the rectangle $$R = \{(t,u) | |t-t_0|\leq T, |u(t)-u_0|\leq L\} $$ and that $$|f(t,u)|\leq M \text{ if } (t,u)\in R$$ Let $\delta = ...
2
votes
1answer
46 views

Solving Lagrange equation.

Please help me solving the equation. I found one of its solutions: $c_1=(xz)/y$ But another one is given as $c_2=(x^3/y)+x$ in the text book. But I cont find. Thank you.
2
votes
1answer
77 views

find the general integrals of the given P.D.E

I tried to find the general integrals of the given P.D.E in the yellow box. And I found $c_1$. But I cannot find another one, say $c_2$. Please help me finding $c_2$. Thank you.
1
vote
1answer
263 views

Find the integral curves of the equation

Question: Find the integral curves of the equation: $$\frac{dx}{y^2x-2x^4}=\frac{dy}{2y^4-x^3y}=\frac{dz}{2z(x^3-y^3)}$$ I could not find any similar example to understand this type of questions ...
2
votes
2answers
55 views

How to calculate Frenet-Serret equations

How to calculate Frenet-Serret equations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), ...
2
votes
1answer
64 views

solve this equation $z(z+y)dx+z(z+x)dy=0$

I need to solve this following equation $$z(z+y)dx+z(z+x)dy=0$$ I get this from above equation $$\frac{dx}{z(z+x)}+\frac{dy}{z(z+y)}=0$$ After there, I dont know what I need to do.
0
votes
1answer
61 views

Find solution(primitive) of the equation

I want to find its solution of the following equation $$ydx+xdy+2zdz=0$$ answer: Keeping $z$ constant; I obtain that $$ydx+xdy=0$$ or $$\frac{dx}{x}+\frac{dy}{y}=0$$ Then I get $$U(x,y,z)=xy$$ ...
1
vote
1answer
55 views

How to obtain the last ratio $\frac{d(x+z)}{(x+z)}$

I am studying example-2.3 In the first line, it says that "the numerators and denominators in the first and last ratio" And the following is obtained $$\frac{d(x+z)}{x+z}=\frac{dy}{y}$$ But I ...
1
vote
2answers
263 views

The Decision of three methods of the solutions $dx/P=dy/Q=dz/R$

Question: (A) $$\frac{adx}{(b-c)yz}= \frac{bdy}{(c-a)xz}=\frac{cdz}{(a-b)xy}$$ (B) $$\frac{dx}{xz-y}=\frac{dy}{yz-x}=\frac{dz}{1-z^2}$$ These are simultaneous diff eq. of the first order and the ...
0
votes
2answers
668 views

Questions about uncoupling dynamical systems and phase plane portraits of the uncoupled systems.

I have found all equilibria, studied their nature, and have been able to make a parametric plot of the following non-linear system along a time axis: $$r'(t)=i-l.r(t)-\text{ux}. r(t). x(t)-\text{uy}. ...
3
votes
1answer
183 views

Constructing a non-linear system with prerequisites about the nature of its critical points.

An exercise from the book I am reading is: "Construct a non-linear system that has four critical points:two saddle points, one stable focus, and one unstable focus." I have tried many systems. I ...
2
votes
2answers
137 views

How to solve this “simple” ordinary differential equation?

I am trying to learn more about calculus by myself, in order to be able to use dynamical systems analysis methods. In a book example, I have to find $f(t)$ from this: ...
4
votes
1answer
310 views

Looking for a logically coherent book for the self-study of differential equations

I'm looking for a logically coherent book for the self-study of differential equations. Let me clarify. By logically coherent, I don't mean proofs of the limit laws, uniqueness theorems etc. By ...
4
votes
1answer
341 views

Which branch of mathematics is this and what are the introductory references?

I am self-studying a physics textbook on waves. While discussing solutions to linear homogeneous ODEs, the author talked about the exponential as "irreducible" solutions and on a footnote, said that ...