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Please ,how to find that the general solution of $u'''(t)=e(t) , t\in [0,1]$ is given by $u(t)=c_0+c_1t+c_2 t^2 +\frac12 \int_0^t (t-s)^2 e(s) ds$

$e:(0,1)\rightarrow \mathbb{R}$, and $e\in L(0,1)$.

i think that it is general homogeneous solution + particular solution ,

the general homogeneous is $c_0+c_1 t+ c_2 t^2$ , but i dont know how to finde that the paricular solution is $\frac12 \int_0^t (t-s)^2 e(s)ds $

Please;

Thank you

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3 Answers 3

You can find the particular solution by doing an integration by parts.

$\frac12 \int_0^t (t-s)^2 e(s)ds = [0-0] -2* \frac12 \int_0^t (t-s) E(s)ds $ where E(s) is a primitive of $e(s)$ such that $E(0)=0$

Similarly we have : $-\int_0^t (t-s) E(s)ds = - ( [0-0] - \int_0^t F(s)ds)$ where $F(s)$...

And u(t)=$\int_0^t F(s)ds$ is a solution of $u'''(t)=e(t)$

EDIT, to be clearer :

u(t)=$\int_0^t u'(s)ds + u(0)$

u(t)=$-\int_0^t (t-s) u''(s)ds + u'(0)t + u(0)$

$u(t)=\frac12 \int_0^t (t-s)^2 u'''(s) ds + u''(0)/2*t^2 + u'(0)t + u(0) = \frac12 \int_0^t (t-s)^2 e(s) ds + u''(0)/2*t^2 + u'(0)t + u(0) $

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But if i dont know that the particular solution is $\frac12 \int_0^t (t-s)^2 e(s)ds$ , how to do ? –  Vrouvrou May 22 '13 at 19:19
    
please @gvo thank you –  Vrouvrou May 22 '13 at 19:26
    
You can use integration by parts from u(t) and replace F with u', E with u'' and then e with u''' in my equations, and go from the last one to the first. The choice of (t-s) as a primitive of 1 would come from fact that it worth 0 when s=t. –  gvo May 22 '13 at 22:02
    
in the last line the integral is from 0 to t. please what i must write in the beginig : $\displaystyle u(t)=\int_0^t \int_0^{\alpha}\int_0^{\beta} u'''(s) ds d{\beta}d{\alpha}$ ?? thank you –  Vrouvrou May 23 '13 at 15:28
    
Corrected, bad copy paste. I don't really understand your question, but it seems related to the answer of anon. –  gvo May 24 '13 at 9:54

It suffices to see that the strange-looking integral is a particular solution. You probably came up with the following type of expression, or at least you should agree it is an obvious way to go:

$$u_p(t)=\int_0^t\int_0^v\int_0^u e(s)dsdudv.\tag{1}$$

Now this is equal to

$$\int_0^te(s)\cdot{\rm Area}(\{(u,v):s\le u\le v\le t\})ds=\int_0^t\frac{(t-s)^2}{2}e(s)ds. \tag{2}$$

How did we get this? First off, the region of points $(u,v)$ such that $s\le u\le v\le t$ (where $s$ and $t$ are fixed) is a right triangle (try plotting some examples to see this) with sides each of length $t-s$, so its area is $\frac{1}{2}(t-s)^2$. But the more substantial formula is the following:

$$\iint\cdots\int_D f(x_0)dx_0dx_1\cdots dx_n=\int f(x_0)\cdot{\rm Vol}(\{(x_0,x_1,\cdots,x_n)\in D\})dx_0 \tag{3}$$

(under suitable hypotheses most likely). This is a "continuous" generalization of a discrete version

$$\sum_{(x,y)\in A}f(x)=\sum_{x\in X}f(x)\,\#\{(x,y)\in A\} \tag{4}$$

(Where $A\subseteq X\times Y$.)

Anyway I presume most of the above is not relevant to you. If you want a way to derive $(2)$ from scratch without knowing ahead of time what to look for, you will need to familiarize yourself with a technique to change double integrals like $\int_0^v\int_0^u e(s)dsdu$ into single integrals (using by-parts integration), and apply it twice to go from $(1)$ to $(2)$.

If you just want to check that $(2)$ is a particular solution, then you can straight-up differentiate the given function three times and check that the result is $e(t)$, using the general formula

$$\frac{d}{dt}\int_0^t f(t,s)ds=f(t,t)+\int_0^tf_t(t,s)ds\tag{5}$$

(and this formula can be derived using the chain rule + fundamental theorem of calculus).


Going from $(1)$ to $(2)$ with by-parts integration: Alright, first let's look at

$$\int_0^v\int_0^u e(s)dsdu. \tag{6}$$

Use by-parts ($X=\int_0^u e(s)ds$ and $Y=u$) to get

$$\int_0^v XdY=[XY]_0^v-\int_0^v YdX=v\int_0^v e(s)ds-\int_0^v ue(u)du=\int_0^v(v-s)e(s)ds. \tag{7}$$

Thus (using by-parts again)

$$\int_0^t\int_0^v\int_0^ue(s)dsdudv=\int_0^t\int_0^v(v-s)e(s)dsdv \tag{8}$$

$$=\int_0^tv\int_0^ve(s)dsdv-\int_0^t\int_0^vse(s)dsdv \tag{9}$$

($dY=vdv$ and $X=\int_0^ve(s)ds$ in the first integral, same by-parts as $(6)$-$(7)$ in second integral)

$$=\left[\frac{t^2}{2}\int_0^te(s)ds-\int_0^t\frac{v^2}{2}e(v)dv\right]-\left[\int_0^t (t-s)se(s)ds\right] \tag{10}$$

$$=\int_0^t\frac{t^2-s^2-2(t-s)s}{2}e(s)ds=\int_0^t\frac{(t-s)^2}{2}e(s)ds. \tag{11}$$


Going from $(1)$ to $(2)$ with reparametrization: The region of integration in ${\bf R}^3$ is

$$D=\{(s,u,v):0\le s\le u\le v\le t\}. \tag{12}$$

Therefore

$$\int_0^t\int_0^v\int_0^ue(s)dsdudv=\iiint_D e(s)dV=\int_0^t\int_s^t\int_u^te(s)dvduds \tag{13}$$

$$=\int_0^t e(s) \left(\int_s^t\int_u^t 1dvdu\right)ds=\int_0^t\frac{(t-s)^2}{2}e(s)ds. \tag{14}$$


Simply checking that the integral expression is a particular solution: differentiating once,

$$\frac{d}{dt}\int_0^t\frac{(t-s)^2}{2}e(s)ds=\frac{(t-t)^2}{2}e(t)+\int_0^t(t-s)e(s)ds. \tag{15}$$

Differentiating a second time,

$$\frac{d}{dt}\int_0^t(t-s)e(s)ds=(t-t)e(t)+\int_0^te(s)ds. \tag{16}$$

Differentiating a third time,

$$\frac{d}{dt}\int_0^te(s)ds=e(t). \tag{17}$$

Hence $u(t)=\int_0^t\frac{(t-s)^2}{2}e(s)ds$ satisfies $u'''(t)=e(t)$.


Proof of $(5)$: let $G(u,t)$ be such that $\frac{dG}{dt}(u,t)=f(u,t)$. Then

$$\frac{d}{dt}\int_0^tf(t,s)ds=\frac{d}{dt}G(t,t)=\frac{dG}{d\,{\small\rm 1st\,coord}}(t,t)+\frac{dG}{d\,{\small\rm 2nd\,coord}}(t,t) \tag{18}$$

$$=\int_0^tf_u(u,s)|_{u=t}ds+f(u,t)|_{u=t}=\int_0^tf_t(t,s)ds+f(t,t). \tag{19}$$

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Nice, I really like this geometrical way to solve this integral. –  gvo May 24 '13 at 9:49
    
@gvo please do you know how to passe from (1) to (2) ? –  Vrouvrou May 24 '13 at 11:43
    
We use Fubini to obtein (2) ??? please –  Vrouvrou May 24 '13 at 16:22
    
@Vrouvrou I have updated my answer with how to go from (1) to (2) using repeated by-parts integration and reparametrization, as well as with how to differentiate (2) three times and get e(t) hence proving it is a particular solution. –  anon May 24 '13 at 19:27
    
thank you , i just don't understand (7) what is dX please –  Vrouvrou May 24 '13 at 20:56

The particular solution

$$u(t)=\frac{1}{2}\int^{1}_{0}(t-s)^2e(s)ds $$

is the Green's function solution $G(t,s)$ for this problem.

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ok ,and so what is the relation with my problem ? , please thank you –  Vrouvrou May 22 '13 at 19:48

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