# Consider the initial value problem

• May 4th 2010, 03:06 PM
taro
Consider the initial value problem
Hi everyone,
This is final time, so I am reviewing all chapters ready for final exam. I get 1 problem that I do not know how to figure out. Could you guys please help. I appreciate it.

Problem:

Consider the initial value problem: y' = 5.y^(4/5) ; y(1) = 0

Consider the two functions f1: R --> R and f2: R --> R such that
f1(x) ≡ 0 , f2(x) = (x -1)^5

a/ Show that f1 and f2 are both solutions to initial value problem.

b/ From part a, we see that the initial value problem has more than one solution. why does this not violate the statement of local existence and uniqueness theorem?
• May 4th 2010, 03:37 PM
dwsmith
$\displaystyle \int y^{\frac{-4}{5}}dy=5\int dx \rightarrow 5y^{\frac{1}{5}}=5[x+c_1]\rightarrow y=(x+c_1)^5$

$\displaystyle y(1)=(1+c_1)^5=0\rightarrow 1+c_1=0\rightarrow c_1=-1$

$\displaystyle y=(x-1)^5$

I am struggling with obtaining y=0; however, Maple found it no problem.
• May 4th 2010, 04:31 PM
taro
here is y' = 5y^(4/5) not y' = 5y^(-4/5)

anyways, thanks dwsmith
• May 4th 2010, 04:36 PM
dwsmith
Quote:

Originally Posted by taro
here is y' = 5y^(4/5) not y' = 5y^(-4/5)

anyways, thanks dwsmith

I used separation of variables that is why I have a negative. Made a mistake with the constant 5 though.

$\displaystyle \frac{dy}{dx}=5y^{\frac{4}{5}}\rightarrow\frac{1}{ y^{\frac{4}{5}}}\frac{dy}{dx}=\frac{1}{y^{\frac{4} {5}}}5y^{\frac{4}{5}}\rightarrow\frac{dy}{y^{\frac {4}{5}}}=5dx$

$\displaystyle \int y^{\frac{-4}{5}}=5\int dx$ because $\displaystyle \frac{1}{y^{\frac{4}{5}}}=y^\frac{-4}{5}$

I am going to fix my original post due to the error with the 5.
• May 4th 2010, 05:11 PM
taro
Oh, I understand now. Thanks alot
• May 4th 2010, 05:24 PM
taro
Sorry, I got a typing mistake. The right problem is:

Quote:

Problem:

Consider the initial value problem: y' = 5.y^(4/5) ; y(1) = 0

Consider the two functions f1: R --> R and f2: R --> R such that
f1(x) ≡ 0 , f2(x) = (x -1)^5

a/ Show that f1 and f2 are both solutions to initial value problem.

b/ From part a, we see that the initial value problem has more than one solution. why does this not violate the statement of local existence and uniqueness theorem?
f1(x) ≡ 0 not f1(x) = 0
• May 4th 2010, 05:25 PM
dwsmith
Quote:

Originally Posted by taro
Sorry, I got a typing mistake. The right problem is:

f1(x) ≡ 0 not f1(x) = 0

Modulo what?
• May 5th 2010, 08:07 PM
chisigma
Quote:

Originally Posted by taro
Hi everyone,
This is final time, so I am reviewing all chapters ready for final exam. I get 1 problem that I do not know how to figure out. Could you guys please help. I appreciate it.

Problem:

Consider the initial value problem: y' = 5.y^(4/5) ; y(1) = 0

Consider the two functions f1: R --> R and f2: R --> R such that
f1(x) ≡ 0 , f2(x) = (x -1)^5

a/ Show that f1 and f2 are both solutions to initial value problem.

b/ From part a, we see that the initial value problem has more than one solution. why does this not violate the statement of local existence and uniqueness theorem?

Writing the ODE in 'standard form'...

$\displaystyle y^{'} = f(x,y)$ , $\displaystyle y(x_{0})=y_{0}$ (1)

... it has one and anly one solution if $\displaystyle f(x,y)$ is 'Lipschitz continous' in $\displaystyle (x_{0},y_{0})$. In Your case is $\displaystyle f(x,y)=\frac{1}{2}\cdot y^{\frac{4}{5}}$ which is not Lipschitz continous in $\displaystyle y_{0}=0$ because its partial derivative is unbounded. Furthemore in this case $\displaystyle f(x,y)$ is function of the y alone and that means that if $\displaystyle \phi (x)$ is solution of...

$\displaystyle y^{'} = \frac{1}{2}\cdot y^{\frac{4}{5}}$ (2)

... the $\displaystyle \phi(x-\xi)$ with $\displaystyle \xi >0$ arbitrary is also solution. In this case...

$\displaystyle \phi(x)=\left\{\begin{array}{ll}(x-1)^{5} ,\,\,x > 1\\{}\\0 ,\,\, x\le 1\end{array}\right.$ (3)

... satisfies (2) with initial condition $\displaystyle y(1)=0$ and that means that $\displaystyle \phi (x-\xi)$ with $\displaystyle \xi >0$ arbitrary also satisfies (2) with the same initial condition...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• May 6th 2010, 03:00 AM
HallsofIvy
Quote:

Originally Posted by taro
Hi everyone,
This is final time, so I am reviewing all chapters ready for final exam. I get 1 problem that I do not know how to figure out. Could you guys please help. I appreciate it.

Problem:

Consider the initial value problem: y' = 5.y^(4/5) ; y(1) = 0

Consider the two functions f1: R --> R and f2: R --> R such that
f1(x) ≡ 0 , f2(x) = (x -1)^5

a/ Show that f1 and f2 are both solutions to initial value problem.

To "show" that a function satisfies a d.e. it is NOT necessary to integrate the equation. If y= f1 is identically 0 then, of course, it is constant and so y'= 0. The equation becomes 0= 5(0)^(4/5)= 0 which is true, and the initial condition, y(1)= f1(1)= 0 is true.
If y= f2= (x- 1)^5, then y'= 5(x- 1)^(4)= 5((x- 1)^5)(4/5) is true and y(1)= (1- 1)^5= 0 is true.

Quote:

b/ From part a, we see that the initial value problem has more than one solution. why does this not violate the statement of local existence and uniqueness theorem?
Well, what exactly does the "local existence and uniqueness theorem" say? Especially the hypotheses. Note that, with y'= F(x,y)= 5y^(4/5), F_y= 4y^{-1/5} which is not defined at x= 0.
• May 6th 2010, 08:57 AM
taro
Thank you so much. I got it.