It is currently Fri Jan 18, 2019 4:29 am

 All times are UTC [ DST ]

 Page 1 of 1 [ 3 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Differential equationsPosted: Thu Jul 14, 2016 10:34 am
 Administrator

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 838
Location: Larisa
Solve the differential equation:

$$f''(x)+5f'(x)+6f(x)=0$$

where $f'(0)=3, \;\; f(0)=2$

and the differential equation:

$$f''(x)+f(x)=\sin 2x$$

where $f'(0)=1, \;\; f(0)=2$

_________________
Imagination is much more important than knowledge.

Top

 Post subject: Re: Differential equationsPosted: Thu Jul 14, 2016 10:35 am
 Team Member

Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
Hello Tolis.

The characterstic polynomial of the first equation is the $\displaystyle{f(y)=y^2+5\,y+6\,y\in\mathbb{C}_[y]}$ with

$\displaystyle{f(y)=0\iff y^2+5\,y+6=0\iff \left(y+2\right)\,\left(y+3\right)=0\iff y=-3\,\,\lor\,\,y=-2}$ .

So, the general solution of tthe equation is $\displaystyle{f(x)=c_1\,e^{-3\,x}+c_2\,e^{-2\,x}\,,x\in\mathbb{R}}$ .

Since $\displaystyle{f(0)=2}$, we get : $\displaystyle{c_1+c_2=2}$ . Also,

$\displaystyle{f^\prime(x)=-3\,c_1\,e^{-3\,x}-2\,c_2\,e^{-2\,x}\,,x\in\mathbb{R}}$ and then :

$\displaystyle{f^\prime(0)=3\iff -3\,c_1-2\,c_2=3\iff -3\,(2-c_2)-2\,c_2=3\iff c_2=9\implies c_1=-7}$, so :

$\displaystyle{f:\mathbb{R}\longrightarrow \mathbb{R}\,,f(x)=-7\,e^{-3\,x}+9\,e^{-2\,x}}$ .

Now, for the second equation :

Consider the equation $\displaystyle{f^{\prime \prime}(x)+f(x)=0}$ with characteristic polynomial

$\displaystyle{g(y)=y^2+1}$ and $\displaystyle{g(y)=0\iff y=\pm i}$, so :

the general solution of the second equation is

$\displaystyle{f(x)=h(x)+k_1\,\cos\,x+k_2\,\sin\,x\,,x\in\mathbb{R}}$ , where

$\displaystyle{h}$ is a partial solution of $\displaystyle{f^{\prime \prime}(x)+f(x)=\sin\,(2\,x)}$ given by :

$\displaystyle{h(x)=\int_{0}^{x}\dfrac{\cos\,t\,\sin\,x-\sin\,t\,\cos\,x}{\cos^2\,t+\sin^2\,t}\cdot \sin\,(2\,t)\,\mathrm{d}t=...=\dfrac{1}{3}\,\sin\,x-\dfrac{1}{3}\,\sin\,(2\,x)\,,x\in\mathbb{R}}$ .

Top

 Post subject: Re: Differential equationsPosted: Mon May 07, 2018 3:08 pm

Joined: Mon May 07, 2018 3:07 pm
Posts: 4
A fairly detailed example, now I have to repeat a complex analysis

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 3 posts ]

 All times are UTC [ DST ]

#### Mathimatikoi Online

Users browsing this forum: No registered users and 0 guests

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net