Solving the Differential Equation for Unique Solution in Region

[Lancelot59]

The given problem for multiple differential equations (not in a system) is: "Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x₀,y₀) in the region."

I'm not entirely sure what to do with this. Looking at the function:

$$\frac{dy}{dx}=y^{\frac{2}{3}}$$

I found the unknown function to be:
$$y=(\frac{x}{3})^3$$
And I have verified that it is a solution.

Now the books says the answer is: "half-planes defined by either y>0 or y<0"
I'm not sure how this solution works. Could someone please explain it to me?

I also don't understand why y=0 isn't valid. I can see y=0 satisfying this equation:

0=0

 

[Dick]

Yes, y=0 is a solution that passes through (0,0). y=(x/3)^3 is also a solution that passes through (0,0). So there is no unique solution if the initial point (x0,y0)=(0,0). What about (x0,y0) being a point besides (0,0)? I thinks that's what they are asking.

 

[lanedance]

the DE is separable
$$\frac{dy}{dx}=y^{\frac{2}{3}}$$

integrating both sides gives
$$\int dyy^{-\frac{2}{3}}=\int dx$$
$$\frac{1}{3}y^{\frac{1}{3}}=x+c$$
$$\frac{1}{3}y^{\frac{1}{3}}=(\frac{x+c}{3})^3$$

this is a family of solutions given by c

now consider whether the following is a solution:
$for \ \ \ \ x \leq 0, \ \ \ \ \ \ y(x) = (\frac{x}{3})^3$
$for \ \ 0< x \leq 1, \ \ y(x) = 0$
$for \ \ \ \ 1< x, \ \ \ \ \ \ y(x) = (\frac{x-1}{3})^3$

 

[Lancelot59]

the DE is separable
$$\frac{dy}{dx}=y^{\frac{2}{3}}$$

integrating both sides gives
$$\int dyy^{-\frac{2}{3}}=\int dx$$
$$\frac{1}{3}y^{\frac{1}{3}}=x+c$$
$$\frac{1}{3}y^{\frac{1}{3}}=(\frac{x+c}{3})^3$$

this is a family of solutions given by c

now consider whether the following is a solution:
$for \ \ \ \ x \leq 0, \ \ \ \ \ \ y(x) = (\frac{x}{3})^3$
$for \ \ 0< x \leq 1, \ \ y(x) = 0$
$for \ \ \ \ 1< x, \ \ \ \ \ \ y(x) = (\frac{x-1}{3})^3$

So you are only using solutions which do not make the equation equal to zero? Is the term homogenous applicable to that case?

 

[HallsofIvy]

On the contrary, Lancelot59's solution explicitely includes y= 0. That's why he can construct an infinite number of solutions such that y(0)= 0.

The "fundamental existence and uniqueness theorem for first order differential equations" says that the problem dy/dx= f(x,y), y(x0)= y0 has a unique solution if there is some neighborhood about (x0,y0) such that f(x,y) is continuous and f(x,y) is "Lipschitz" in y ("differentiable with respect to y is a simpler, sufficient but not necessary condition) in that neighborhood.

Here, $f(x,y)= y^{2/3}$ is continuous in any neighborhood of (0,0) but $f_y= (2/3)y^{-1/3}$ does not exist at (0, 0) so the function is not Lipschtz there.

We can guarantee a unique solution on any neighborhood that does NOT include the x-axis but not in any neighborhood that does.

 

[Lancelot59]

On the contrary, Lancelot59's solution explicitely includes y= 0. That's why he can construct an infinite number of solutions such that y(0)= 0.

The "fundamental existence and uniqueness theorem for first order differential equations" says that the problem dy/dx= f(x,y), y(x0)= y0 has a unique solution if there is some neighborhood about (x0,y0) such that f(x,y) is continuous and f(x,y) is "Lipschitz" in y ("differentiable with respect to y is a simpler, sufficient but not necessary condition) in that neighborhood.

Here, $f(x,y)= y^{2/3}$ is continuous in any neighborhood of (0,0) but $f_y= (2/3)y^{-1/3}$ does not exist at (0, 0) so the function is not Lipschtz there.

We can guarantee a unique solution on any neighborhood that does NOT include the x-axis but not in any neighborhood that does.

I see, so the trick is to differentiate the original differential equation with respect to the "dependent" (Y in this case) variable, and see where solutions for it are valid?