### Author Topic: Integral curves  (Read 1111 times)

#### Victor Ivrii

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##### Integral curves
« on: January 09, 2018, 03:38:46 AM »
Someone claimed "I took MAT244 but never heard about integral curves". LOL, on the bottom of page 12 of Boyce-DiPrima 10E
Quote
The geometrical representation of the general solution (17) is an infinite family of curves called integral curves. Each integral curve is associated with a particular value of c and is the graph...
after which integral curve or integral curves is repeated more than 30 times.

#### Jingxuan Zhang

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##### Re: Integral curves
« Reply #1 on: January 09, 2018, 10:05:37 AM »
Please confirm with me if the following is correct. Suppose we have
$$a(x,y)u_{x}+b(x,y)u_{y}=f.$$
Then our LS expression is the directional derivative of $u$ on
$$\frac{dx}{a(x,y)}=\frac{dy}{b(x,y)}.$$
But how are we to read the RS expression under this context? Is it something related to this field? Suppose $f=0$, then it means $u$ is constant along each integral curve of the above field. But what if $f$ is some other function?

#### Victor Ivrii

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##### Re: Integral curves
« Reply #2 on: January 09, 2018, 10:29:36 AM »
$$\frac{dx}{a(x,y)}=\frac{dy}{b(x,y)}=\frac{du}{f(x,y,u)}$$
so if integral curve is parametrized by $\tau$: $dx= a\,d\tau$, $dy=b\,d\tau$ then along this curve $du=f\,d\tau$, we get ODE along integral curve.

Examples of variable coefficients will be considered during tutorial and  the next lecture.
« Last Edit: January 10, 2018, 11:05:12 AM by Victor Ivrii »

#### Ioana Nedelcu

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##### Re: Integral curves
« Reply #3 on: January 09, 2018, 11:03:21 PM »
I'm pretty sure the "integral curves" refer to the representation of the solution in the direction field