38
votes
15answers
6k views

Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background. One question is haunting me. Why do I need ...
3
votes
1answer
93 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
72 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
85 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
67 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
41 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
34 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
55 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
48 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
77 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
64 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
179 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
86 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
54 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
367 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
56 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
78 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
300 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
63 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
67 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
64 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
57 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
294 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
788 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
191 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
142 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
389 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
351 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 ...