Solving an Initial Value Problem with Separable Differential Equations

[jjr]

Homework Statement

The problem is from Walter Gautschi - Numerical Analysis, exercise 5.1.

Consider the initial value problem

$\frac{dy}{dx}=\kappa(y+y^3), 0\leq x\leq1; y(0)=s$

where $\kappa > 0$ (in fact, $\kappa >> 1$) and s > 0. Under what conditions on s does the solution $y(x) = y(x;s)$ exist on the whole interval [0,1]? {Hint: find y explicitly.}

The Attempt at a Solution

Following the hint, I tried solving it as a separable differential equation:

$\frac{dy}{y+y^3} = \kappa dx$

(Using wolframalpha here)

$log(y) - \frac{1}{2}log(y^2+1) = \kappa x$

$10^{log(y)-\frac{1}{2}log(y^2+1)} = 10^{\kappa x}$

Ending up with

$\frac{y}{2} + \frac{1}{2y} = 10^{-\kappa x}$

Not sure how to solve this, so used wolframalpha again and got:

$y = 10^{-\kappa x} \pm \sqrt{10^{-2 \kappa x} - 1}$

Evidently $y(0) = 1$, which means that s has to be equal to 1.

This doesn't seem right to me. Especially considering the work needed to find y explicitly, seeing as how this is not a course in differential equations. So what I am missing here?

 

[pasmith]

Homework Statement

The problem is from Walter Gautschi - Numerical Analysis, exercise 5.1.

Consider the initial value problem

$\frac{dy}{dx}=\kappa(y+y^3), 0\leq x\leq1; y(0)=s$

where $\kappa > 0$ (in fact, $\kappa >> 1$) and s > 0. Under what conditions on s does the solution $y(x) = y(x;s)$ exist on the whole interval [0,1]? {Hint: find y explicitly.}

The Attempt at a Solution

Following the hint, I tried solving it as a separable differential equation:

$\frac{dy}{y+y^3} = \kappa dx$

(Using wolframalpha here)

$log(y) - \frac{1}{2}log(y^2+1) = \kappa x$

I see no constant of integration; the right hand side should be $\kappa x + C$.

 

[jjr]

Thank you, and right you are; that was quite sloppy of me. So starting then at

$\frac{y}{2}+\frac{1}{2y}=C10^{\kappa x}$

yielding

$y = C10^{-\kappa x} \pm \sqrt{ C^{2}10^{-2 \kappa x} - 1 }$

so

$y(0) = s = C \pm \sqrt{C^2-1}$

Which imposes a constraint on C: $|C| \geq 1$

But they are asking for what conditions on s the solution $y(x) = y(x;s)$ exists on the interval [0,1]? I am having some trouble seeing the connection here, especially in the context of the chapter "Initial Value Problems for ODEs: One-Step Methods". Any thoughts?

 

[pasmith]

Thank you, and right you are; that was quite sloppy of me. So starting then at

$\frac{y}{2}+\frac{1}{2y}=C10^{\kappa x}$

$$\log(y) - \frac12 \log(y^2 + 1) = \log\left( \frac{y}{\sqrt{1 + y^2}}\right).$$ This is not in general equal to $\log(y/2 + 1/(2y))$, which is what your expression implies (and I must apologise for not pointing this out in my first reply).

Also: in calculus, "log" means "natural logarithm", so if $a = \log b$ then $b = e^a$.

 

[jjr]

Ah, that explains the seemingly frivolous interchangeable use of log and ln I've sometimes seen. And yes, I see that I've made en error when dealing with the $\frac{1}{2}$ factor. Well, being more explicit this time and starting from (I will for my own sake switch to using ln now):

$ln(y) - \frac{1}{2}ln(y^2+1) = \kappa x + C_1$

$ln(y) - ln( \sqrt{y^2+1}) = \kappa x + C_1$

$e^{ln(y) - ln(\sqrt{y^2+1})} = C_2 e^{\kappa x}$

$\frac{e^{ln(y)}}{e^{ln(\sqrt{y^2+1})}} = C_2 e^{\kappa x}$

$\frac{y}{\sqrt{(y^2+1)}} = C_2 e^{\kappa x}$

$\frac{y^2}{y^2+1} = C_3 e^{2\kappa x}$

$\frac{y^2+1}{y^2} = C_4 e^{-2\kappa x}$

$1 + y^{-2} = C_4 e^{-2\kappa x}$

$y^{-2} = C_4 e^{-2\kappa x} - 1$

$y^2 = \frac{1}{C_4 e^{-2\kappa x} -1}$

$y = \pm \frac{1}{\sqrt{C_4 e^{-2\kappa x} -1}}$

So

$y(0) = s = \pm \frac{1}{\sqrt{C_4 - 1}}$

and thus $C_4 > 1$.

Hopefully the math checks out this time. But I am still not sure how to use this in the context of the exercise. Any ideas?

 

[ehild]

It was given that s>0. Find the integration constant C4 in terms of s. Than write y(x) using s, and see what should be true for s, so y(x,s) is defined for the whole interval [0,1].

ehild

 

[jjr]

Perfect, thanks!

 

[ehild]

You are welcome

ehild