MAT244--2020F > Chapter 2

Boyce-DiPrima Section 2.1 Example 1

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jimboxie:
I'm a bit rusty on some calc, and I was wondering how the textbook did the following, along with my attempt on what I think happened (image attached), but i can't seem to see what I did wrong.

Suheng Yao:
I think that the textbook just uses the product rule: [f*g]=f'*g+g'*f. Here, you can think f as 4+t^2 and g as y. Hope this helps.

jimboxie:
I completely forgot about the product rule. Thank you.

RunboZhang:
I think Suheng Yao has pointed out a crucial point: product rule. Indeed, if you observe LHS=(4+t^2)(dy/dt) + 2ty, you may find the pattern of u'v+uv' where u=y and v=4+t^2. By applying product rule here, we can make LHS become a single expression and thus make it a separable differential equation (which is already discussed in the lecture).

I think a general method of solving this kind of problem is introduced later in the textbook and it is quite useful in solving inseparable first order differential equation.

Victor Ivrii:

--- Quote from: Suheng Yao on September 15, 2020, 02:51:19 PM ---I think that the textbook just uses the product rule: [f*g]=f'*g+g'*f. Here, you can think f as 4+t^2 and g as y. Hope this helps.

--- End quote ---

Please, never use * as multiplication sign (which even is not needed here). Your formulas are correctly formatted, but if you surround each by dollar sign like

--- Code: ---I think that the textbook just uses the product rule: $[f*g]=f'*g+g'*f$. Here, you can think $f$ as $4+t^2$ and $g$ as $y$. Hope this helps.
--- End code ---
you'll get
I think that the textbook just uses the product rule: $[f*g]=f'*g+g'*f$. Here, you can think $f$ as $4+t^2$ and $g$ as $y$. Hope this helps.

Again, * is reserved for a different operation, and it usage as multiplication sign may be considered as a mathematical error